Transcript Slide 1

9.2 Spac
e
1. Gravitational fields
Weight
Weight is the force on an object due to a
gravitational field.
The rest mass of an object remains
constant irrespective of the gravitational
field.
The weight of an object changes as the
gravitational field changes:
FW = mg
Where,FW = weight (N)
m= mass (kg)
g= gravitational acceleration (ms-2)
The mass of an astronaut at rest on
Earth remains the same on another
planet
The weight of an astronaut on another
planet will depend on the planet's
gravitational field
Calculating weight
The reading on the scale is 57 kg.
Although the scale actually
measures weight (it measures a
force), scales are traditionally
calibrated in the units for mass.
The weight of the person on the
scales can be calculated using :
FW = mg
FW = weight N; m = 57kg and g= 9.8ms-2
FW = 57 x 9.8 = 558.6 N
The value of g on Venus is 8.9ms-2 .
On Venus, this person would still
have a mass of 57 kg but her weight
would be 447 N (57 x 8.9 )
Gravitational Potential Energy
The change in gravitational potential
energy is related to the work done:
Work: W = Fs (force used to move an
object from the surface to a height of
s metres)
In a gravitational field: F= Fw = mg
Therefore
W = mgs
Potential energy near the Earth's
surface is given by:
Ep = mgh
The two expressions are clearly
identical
W = Ep
Gravitational
Potential
Energy
m1m2
Ep   G
r
G = 6.67300 × 10-11 m3 kg-1 s-2
(Universal gravitation constant)
Gravitational potential energy is the work
done to move an object from a very large
distance away to a specific point in a
gravitational field.
When a rocket is launched into space,
work is done on the rocket hence the
rocket's gravitational energy increases
When a meteor approaches the Earth its
gravitational potential energy decreases
(as the meteor speeds-up, potential energy
is converted into kinetic energy )
The lowest Ep in the gravitational field
surrounding a planet is at the surface of
the planet.
The highest Ep is when that of object is at
an 'infinite' distance from Earth. Its
gravitational energy due to Earth's
gravitational at infinity is zero.
It follows that all other values of Ep must
be negative
Ep example problem
Calculate the energy that was
required to move The Hubble
Telescope (11 110 kg) to its parking
orbit (559 km)
To do this we have to find the
change in energy from its value on
the surface to its value in orbit
Determination of g
To use a simple pendulum to determine the
value of g.
1) The equipment used is shown to the left.
2) Displace the pendulum through a small
angle ( <10o) and set it swinging.
3) Using a stop watch measure the time
taken for ten periods (a period is a full
swing from A to B and back to A)
4) Repeat a number of times and find the
average period (T) for that length (L)
5) Repeat steps 3 to 6 for at least three
different lengths.
6) Plot T2 vs. L.
7) The gradient of the line is: T2 / L
8) Using the formula adjacent and the
gradient of the line, work out a value for
g
Variations from 9.8 ms-2
The pendulum experiment generally yields fairly accurate results , ranging from 9.5
to 10 ms-2 for g.
Experimental errors occur in any experiment. In this case there are errors in
measurement as well as errors inherent in the equipment used.
The average gravitational acceleration at the surface of the Earth is 9.80 ms-2. It can
vary by a total of up to about 1%. This is largely due to a combination of 3 factors:
Latitude:
Gravity is an average 9.78 ms-2 at the equator. It is about 0.5% higher at the poles
at an average 9.83 ms-2. This is due to a combination of effects:
The spin of the Earth creates an outward force that is greatest at the equator. The
difference is small, but enough to make the launch of space rockets cheaper near
the equator than near the poles for most orbits.
The Earth's diameter at the equator is about 12,756 km . The distance from pole to
pole is about 12,713 km This means that the surface at the equator is further
away from the centre of the Earth.
Elevation:
Gravity decreases with distance from the centre of the Earth. Gravity is about
0.2% lower at the top of Mount Everest than at sea level
Crust thickness and Density:
Variations in the density and thickness of rocks in the Earth's lithosphere cause
local variations in gravity. The denser the rock, the higher the value of g.
g for other planets
The gravitational force exerted by a planet at its surface is
related to its mass and its radius.
The radius and mass of a planet can be derived from
astronomical observation.
g for any planet (including Earth) can be estimated using the
formula:
Revision questions - 1
1) Define weight
2) Explain the relationship between a change in gravitational potential
energy and the work done
3) Define gravitational potential energy
4) Describe an investigation you performed to determine a value for the
acceleration due to gravity using pendulum motion
5) Identify reasons for possible variations of g from the value of 9.8 ms-2
6) Identify the data you would need in order to predict the value of g on
other planets and describe how you would use it.
7) Calculate the work required to lift a 35 000 kg rocket to an altitude of 1
200 km
8) An astronaut lands on a new planet where the value of g is 10.67 ms-2. If
the weight of the astronaut on Earth was 796 N determine his weight on
the new planet
2. Rocket Launch
Projectile
motion
Projectile motion is considered as a
motion in two dimensions: horizontal and
vertical.
The horizontal velocity is constant if we
ignore air resistance.
The vertical velocity is changed by the
action of the acceleration due to gravity
“g” which points directly downward at all
times (toward the centre of the Earth or
planet)
The trajectory is parabolic (if we ignore air
resistance).
If the projectile starts and finishes at the
same level, then the time to maximum
height is at ½ of the time of flight.
At maximum height the vertical velocity is
zero
If the initial velocity is upward and is taken
as positive then downward is the negative
direction. Hence g must be negative.
Projectile motion
The two velocities are the
component vectors of the
projectile velocity at any one
time during the flight.
The horizontal component (x
direction) remains constant
throughout.
The vertical motion (y direction)
is uniformly accelerated
downward by -g.
At any time during the flight the
velocity and/or the
displacement of the projectile
can be calculated by finding the
vertical and horizontal
components and then using
Pythagoras and trigonometry
The time of flight is the same
for both motions and is used to
tie them together.
Projectile motion
Horizontal motion is described by:
Where, t = time of flight; x = range
Vertical motion is described by:
Where at maximum height; uy = 0
Projectile motion - a worked example
An object is fired at 100 ms-1 at an angle of 47o near the edge of a cliff face, which is 58 m above sea level. Find
how far from the bottom of the cliff the object will enter the sea.
Step 1 do a sketch of the problem.
Step 2 resolve the initial velocity into the two components ux and uy
ux=100cos47 = 68.1998 ms-1 and uy=100sin47 = 73.1353 ms-1
Step 3 find the time to point H
vy=uy+ayt
0=73.1353 + (-9.8)t
t = 7.4628 s
Step4 find height of point H
vy2 = uy2 +2ayy
0 = (73.1353)2 + 2(-9.8)y
y = 272.8965 m
H from sea level = 272.8965 + 58 = 330.8965
Step 5 find time to impact from point H (y = - 330.8965 m)
y=ut+1/2at2
- 330.8965= 0t + ½ (-9.8)t2
t = 8.2176 s
Step 6 find the total time of flight
t = 7.4628 + 8.2176 = 15.6804 s
NB: the above can be done in one step using the 'quadratic formula' on y=uyt+1/2at2 with y = -58 and uy= 73.1353
Step 7 find the horizontal distance traveled
x=uxt x = 68.1988 x 15.6804
x =1069.38446352 = 1070 m (do not round-off to significant figures until the final answer)
Projectiles – an investigation
An example of an investigation is the construction of a small catapult. You will
need to research the subject and then come-up with your own design.
You will need to calibrate it by firing an object a number of times. Keep a full
record of all your experiments in a log book.
On the side of your catapult you will need to attach some sort of scale that will
allow you to bring back the arm reproducibly at different points to achieve
different ranges.
Find the maximum range for your machine. Then try to set your arm so the it
will propel the object ¾ of the way, ½ way and ¼ of the way. For each position
validate your results by repetition.
For each position estimate angle of flight, maximum height time of flight using
a camcorder or the video recorder on a mobile phone. From these calculate
the initial velocity of the object
Next, set your self a fixed distance to aim at (e.g. 60% of your maximum
range). Set your catapult using your calibrations and check how accurate you
were.
Finally, write a report on your investigation. When you submit it for marking,
include your log book.
Galileo Galilei
Before Galileo, it was believed
that a projectile would travel in a
straight line until it ran out of
“impetus” and then it would fall
straight down.
Using both experiment and
mathematics, Galileo showed
that the path of a projectile is a
parabola.
Galileo rolled an inked bronze
ball down an inclined plane onto
a table. The ball thus accelerated,
rolled over the table-top with
uniform motion and then fell off
the edge of the table. Where it hit
the floor, it left a small mark. The
mark allowed Galileo to measure
the height and horizontal
distance for different velocities.
Galileo Galilei
Using a thought experiment,
Galileo also showed that all
objects will fall under the
influence of gravity at the same
rate irrespective of their masses
(if there is no air resistance)
He didn't actually drop objects
from the leaning tower of Pisa as
legend describes.
Galileo's discovery was
demonstrated by the Apollo
mission astronauts in the
vacuum of the Moon.
They dropped a feather and a
hammer at the same time - as
expected, both the feather and
the hammer hit the lunar surface
exactly at the same time! (A short
clip of this is available on
YouTube)
Escape velocity (speed)
For a given gravitational potential energy, the
escape velocity is the minimum speed an object
(without propulsion) needs to have so that its
kinetic energy equals the gravitational potential
energy.
An object on the surface of a planet will need a
kinetic energy equal to the gravitational potential
energy at the surface of the planet. For the Earth's
surface that kinetic energy results from a speed of
about 11,100 ms-1
The concept does not directly apply to rockets as
they are propelled by their engines and
accelerating. But at the instant the propulsion
stops, the vehicle can only escape if its speed is
greater than or equal to the escape velocity for
that position.
It would apply to an object fired by gun in an
attempt to, say, reach the Moon. Much like Jules
Verne's “moon gun” described in his 1865 novel
From the Earth to the Moon.
Escape velocity is proportional to the square root
of the planet's mass and inversely proportional to
the square root of the distance from its centre,
but is not affected by the object's mass.
Newton's thought experiment related
to firing a gun pointing horizontally,
on top of a very high mountain. At
lower muzzle velocities the cannon
ball would fall to the ground in a
parabolic trajectory as shown by
Galileo.
As the gun's muzzle velocity is
increased the ball will hit further and
further, but eventually, because the
Earth is round it will end up not being
able to hit the ground at all (because,
as it falls, the surface of the Earth
curves away from it.
The ball would complete a circular
orbit around the Earth and if the gun
was moved in time and there was no
air resistance the ball would be in a
circular orbit around the Earth
If the ball was fired at even higher
speeds the orbit would become an
ellipse.
At speeds above escape velocity the
trajectory would become a hyperbola
and the ball would leave forever.
Newton's escape
velocity
G-forces
An acceleration of 1g is equal
to standard gravity (9.80 ms-2)
G force = (a – g) /g
Human tolerances depend on
the magnitude of the G-force,
the length of time it is applied,
the direction it acts, the
location of application, and
the posture of the body
A constant 16 g for a minute
can be Fatal.
Astronauts and test pilots are
tested for their ability to
withstand high G's in special
equipment.
G forces and the Apollo space craft*
*http://history.nasa.gov/ap15fj/01launch_to_earth_orbit.ht
m
1. Launch with ignition of the S-IC. Note how the
acceleration rapidly rises with increasing
engine efficiency and reduced fuel load.
2. Cut-off of the centre engine of the S-IC.
3. Outboard engine cut-off of the S-IC at a peak of
4g.
4. S-II stage ignition. Note the reduced angle of
the graph for although the mass of the first
stage has been discarded, the thrust of the S-II
stage is nearly one tenth of the final S-IC thrust.
5. Cut-off of the centre engine of the S-II.
6. Change in mixture ratio caused by the
operation of the PU valve. The richer mixture
reduces the thrust slightly.
7. Outboard engine cut-off of the S-II at a peak of
approximately 2.7g.
8. S-IVB stage ignition. Note again the reduced
angle of the graph caused by the thrust being
cut by a fifth.
9. With the cut-off of the S-IVB's first burn, the
vehicle is in orbit..
The 4g reached during boost is not the highest that
will be experienced during the mission. Entry
through the Earth's atmosphere decelerates the
Command Module at about 6½g.]
Robert H. Goddard
Robert H Goddard ( 1882 – 1945), was an American physics professor and
scientist. He was a pioneer of controlled, liquid-fueled rocketry.
In 1926 he launched the world's first liquid-fueled rocket and from 1930 to
1935, he launched rockets that attained higher speeds and higher altitudes.
His work in this field was revolutionary, he is recognized as one of the fathers
of modern rocketry.
In 1914, his first two (of 214) patents were accepted and registered. The first,
described a multi-stage rocket. The second, described a rocket fueled with
gasoline and liquid nitrous oxide.
In 1919, the Smithsonian Institution published Goddard's advanced treatment
on rocketry: “A Method of Reaching Extreme Altitudes”. In it he describes his
mathematical theories of rocket flight, his experiments with solid-fuel rockets,
and the possibilities he saw of exploring the earth's atmosphere and beyond.
This book is regarded as one of the pioneering works of the science of
rocketry.
On Sept. 16, 1959, the U.S. Congress authorized the issue of a gold medal in
the honor of Professor Robert H. Goddard
Launching a rocket from Earth
Earth orbits the Sun at a speed of
107,000 kmh-1 Launching a rocket in the
same direction as Earth's movement,
will provide a big boost in relative speed
(to the Sun) and save a lot of fuel.
Earth rotates toward the East, one
complete turn each day. This means
that at the equator, Earth's surface has
a linear speed of about 1600 kmh-1. For
this reason most launch sites are
located near the equator and rockets
are launched toward the east, thus
obtaining a 'free' boost from Earth's
rotational motion.
The eastward launch time is chosen to
give the rocket the time to accelerate as
it goes partway around Earth.
For interplanetary travel, it is important
to chose the right time of the year (i.e.
the position of the Earth in its orbit) to
launch. This is known as a “launchwindow”
Off the pad and into orbit
Rockets can't fly in a vacuum ...
On January 12, 1920 a front-page story in The New York Times editorial delighted in heaping
scorn on the proposal that travel to the Moon was possible. It expressed disbelief that:
“Professor Goddard actually does not know of the relation of action to reaction, and the need
to have something better than a vacuum against which to react" !
Of course the newspaper was completely wrong (and that is not news). The action-reaction
relationship is between the rocket and the gases emitted, and not between the rocket and air.
This is an example of how the Law of Conservation of Momentum, which is Newton's third law,
can be used to analyse rocket motion.
For a rocket at rest, the momentum of the system (rocket + fuel) is zero. According to the Law
of Conservation of Momentum this must be preserved.
The mass of fuel burned / second is much less than the mass of the rocket, but due to the
explosive nature of rocket fuel, the velocity of the hot exhaust gases will be far greater than the
velocity of the rocket
As fuel is burnt the mass of the rocket diminishes but its velocity increases. The momentum of
the system remains zero.
Forces on astronauts
Two forces act on an astronaut during launch: the
thrust (T) and the astronaut's own weight (mg).
Newton’s second law is used to arrive at the
relationship between the acceleration of the rocket and
the forces on the astronaut,
The astronaut experiences this acceleration as an
apparent increase in his/her weight of a magnitude that
depends on the rocket's acceleration. An acceleration
of 4g is experienced as a four-fold increase in weight.
Once the rocket reaches a stable orbit a
'weightlessness' is experienced. The astronauts are
not really weightless but are continuously falling (in
free-fall) toward the centre of the Earth with an
acceleration of g (for that altitude)
During re-entry the spacecraft must be decelerated as
quickly and as safely as possible. The acceleration of g
downward is opposed by air resistance, which at high
speeds can cause the space ship to heat-up to
dangerous temperature.
The re-entry trajectory is long and shallow to decrease
the re-entry g-forces on the astronauts, to decrease the
speed of the aircraft and so avoid burn-up or impact at
a fatally damaging speed.
Circular orbits
Any object that is moving in a circular path at a uniform speed is accelerating toward the centre
of the circular path (its speed is constant, but its direction is constantly changing). The
acceleration is called centripetal (centre-pointing) and is associated with a force called the
centripetal force:
Fc = mac
(m= mass of the object in circular motion)
For circular motion it can be shown that:
ac = v2 / r (v = instantaneous tangential speed; r = radius of circle)
Therefore ,
Fc = mv2 / r
For a rock at the end of a string that is being rotated in uniform circular motion, the Fc is the
tension in the string. For a satellite or planet in uniform circular orbit, the Fc is the gravitational
attraction between the satellite / planet (m) and the central body (M). Hence:
Fc = Fg
mv2 / r = G M m / r2
v2 = G M / r

'v' is the orbital velocity of the satellite, it is independent of its mass, but dependent on the
mass of the central body (planet or star).
Two types of orbits
Low Earth orbits
A low Earth orbit (LEO) is an
orbit that lies between the
Earth’s upper atmosphere and
the lower van Allen radiation
belts. The average altitude for
a LEO is from 200 km to 1,000
km above the surface of the
Earth.
An object in LEO will need an
orbital velocity of
approximately 28.000 km h-1
to maintain the orbit,
Objects in LEO will have
orbital periods of about 90
minutes.
The space shuttle, the Hubble
space telescope and the
International Space-station
are in LEO's
Geostationary orbits
A satellite in a geostationary
orbit (GO) has an orbital
period equal to one sidereal
day (23 hours 56 minutes).
From the ground, a satellite in
GO appears fixed in the sky
because its period is the same
as the Earth's.
A satellite in GO will have an
altitude above the upper van
Allen belts of approximately
35 800 km.
An object in GO will need an
orbital velocity of about 11,000
km h-1.
Satellites in GO are mainly
country- specific
communications and weather
satellites.
What happened to Skylab?
Skylab was a space research laboratory constructed by NASA and was launched
in 1973.
In mid 1979 it re-entered the atmosphere and landed in fiery chunks around the
Balladonia Hotel/Motel (Western Australia). There were no injuries.
Jimmy Carter, the US President at the time, personally rang the Balladonia
Hotel/Motel to apologise to the shaken but not injured staff and guests.
What went wrong?
Through a process called 'orbital decay' satellites in Low Earth Orbits are slowed
down by the friction with the very thin air of the upper atmosphere. This friction
results in a loss kinetic energy.
The satellite drops down to lower altitude and gains kinetic energy at the expense
of gravitational potential energy. It will now be moving faster but a faster velocity
means increased friction. At this lower altitude, the air is even denser. The loss in
kinetic energy now is even greater, so the process picks up speed and the
satellite is soon on a no-return path down to Earth.
Most satellites in LEO are small enough to burn up from air-friction before
reaching the ground. Skylab was just too big (88,900 kg) and pieces of it
managed to hit the ground in WA.
The Hubble Space telescope (11,100 kg) orbits in a LEO too and if it is not reboosted by a shuttle or other means, it will re-enter the Earth's atmosphere
sometime between 2010 and 2032.
Keppler's Law of Periods (3rd)
Notice that,
With a system of planets or satellites rotating a central body, the
right hand side of the equation remains constant and does not
depend on the individual masses of the rotating bodies but only
on the mass of the central body
If the ratio of the (radius)3 to the (period)2 is known for one body,
it will be the same for all the other bodies rotating around the
same central star or planet.
Orbital velocity, the gravitational constant, mass
of the central body, mass of the satellite and the
radius of the orbit
Orbital velocity is the
rectilinear,
instantaneous speed
tangential to the orbital
path in the direction of
movement.
Keppler's law can be
used to find the
relationship between
orbital velocity and
orbital period.
Mass of the satellite and
radius of the orbit can
be related to the period
of the orbit.
Re-entry
For spacecraft needing to re-enter the Earth's atmosphere safely, there is only
a small angular window available.
If the angle is too shallow the spacecraft will bounce on the atmosphere like a
flat stone on a pond of water; if the angle is too steep the spacecraft will suffer
too much drag from the air and will burn up and the deceleration will be too
rapid.
The angles differ for different orbits and different spacecraft: for the Apollo
capsule the window was between 5o and 7o but for the space shuttle orbiter it is
between 1o and 1.5o.
There are other consideration for a safe re-entry and a safe landing.
Spacecraft in orbit have kinetic energy, as the craft descends even more kinetic
energy is produced as it loses gravitational potential energy. On re-entry, some
of this energy is converted into a lot of heat by air friction. The effect of this
heat can be minimized by 2 methods:
By the use of an ablative shield (substances that will burn at a very high
temperature and thus use-up the heat, like on the Apollo capsule underside).
By using a long and gradual decent with a heat- insulating shield on the
underside (the space shuttle uses this method).
Re-entry
The heat generated also causes a general
radio black-out with the craft as the overheated air molecules become ionized. For a
period of time the craft cannot be contacted
and the crew cannot be warned of any
dangerous situations that might be
developing e.g. being off-course
The orbit speeds are in excess of 20 000 km
h-1 to decrease from that speed down to a
tolerable impact velocity in a distance of just
200 or so km would need very high
decelerations (work it out). Hence the re-entry
trajectory has too be very shallow resulting in
a long path, even then the deceleration is
very high and the G-forces on the astronauts
can be considerably higher than launch Gforces.
Safe impact with the surface is achieved by
the use of parachutes deployed at correct
altitudes, by splashing in water rather than on
hard ground or by gliding down to an airport
tarmac like glider (space shuttle)
Revision questions - 2
1) Describe the trajectory of an object undergoing projectile motion
within the Earth’s gravitational field in terms of horizontal and
vertical components
2) Calculate the maximum height, the time of flight and the range of an
object with an initial velocity of 30 ms-1 that is projected at 3 different
angles: 30o; 45o and 60o
3) Describe a first-hand investigation you performed to analyse data
and to calculate initial and final velocity, maximum height reached,
range and time of flight of a projectile.
4) Describe Galileo’s contribution to projectile motion.
5) Explain the concept of escape velocity.
6) Outline Newton’s concept of escape velocity.
7) Identify why the term ‘g forces’ is used to explain the forces acting
on an astronaut during launch
8) Present information on the contribution of one of the following to the
development of space exploration: Tsiolkovsky, Oberth, Goddard,
Esnault-Pelterie, O’Neill or von Braun
Revision questions - 3
1) Discuss the effect of the Earth‘s motion in space on the launch of a
rocket.
2) Describe the changing acceleration of a 2 stage rocket during launch.
3) Describe the forces experienced by astronauts during launch.
4) Describe the forces involved with uniform circular motion.
5) Compare low Earth and Geostationary orbits
6) Account for the orbital decay of satellites in low Earth orbit.
7) Calculate the orbital speed and period of a satellite orbiting at an
altitude of 350 km above the Earth's surface. Explain why this is not a
Geostationary orbit.
8) The average radius of Earth's orbit is 149,600,000 km and its period
of rotation is 365.26 days. If Jupiter rotates the Sun in 4332.71 Earth
days, calculate the average distance of Jupiter from the Sun
9) Define the term orbital velocity and using Keppler's 3rd law find its
relationship to the period of the orbit
10) Outline issues that relate to a safe re-entry and touch down for a
manned spacecraft in parking orbit around the Earth. Identify which
of these would not apply to a safe landing on the Moon's surface
3. Gravity
Gravity
Gravity is the force of attraction
between two or more masses.
Each mass is thought of having an
infinite gravitational field surrounding it
in space
The gravitational force exerted by an
object is proportional to its mass.
The strength of the field obeys the
inverse- square law (it diminishes as the
inverse of the square of the distance
from the centre of mass).
Gravity is the weakest of the four known
forces, but large bodies like the Sun and
the planets exert strong gravitational
fields because of their large masses
Gravity holds the planets and satellites
in their orbits it even stops the
thermonuclear fusion reaction in the
Sun from exploding it apart.
Gravity shapes the structure and
evolution of stars, galaxies, and the
entire universe...
Newton's Law
Newton's Law of Universal
gravitation states”
“Every object in the universe
attracts every other object
with a force directed along
the line joining the centres of
mass of the two objects.
The force point toward each
object and is proportional to
the product of their masses
and inversely proportional to
the distance separating their
centres”.
m1m2
F G 2
d
G = 6.67300 × 10-11 m3 kg-1 s-2 (Universal
gravitation constant)
Newton's law and the motion of
satellites
As previously shown;
Newton's Law of gravitation
is needed to derive an
expression for the orbital
velocity of a satellite in near
circular orbit (it is also used
for elliptical orbits but the
math involved is beyond our
scope).
Newton's Law can also be
used to derive Keppler's
third law.
With these two laws the path
or orbit of any satellite or
planet can be analyzed and
calculated. Predictions can
be made as to its future
positions to the extent that
spacecraft can find them in
the vastness of space.
Slingshot
The slingshot effect is also know as a 'flyby' or as
a 'gravity assist' trajectory. It consists of piloting
or directing the spacecraft close to a planet in
order to achieve a change in the direction or a
change in velocity without the expenditure of fuel.
Depending on the direction of approach of the
spacecraft relative to the planet's motion, a
spacecraft' s velocity can be increased (if the
destination is further on) or decreased (if a
parking orbit is the aim)
In a gravity-assist trajectory, momentum is
transferred from the orbiting planet to a
spacecraft approaching it (it's not unlike an
elastic collision between two billiard balls, except
one of the balls is millions of times more massive
than the other!)
Both gain/lose the same amount of momentum
but the planet shows negligible change in its
velocity as it is so massive, whereas the
spacecraft shows a significant change in it's
speed (and / or direction)
Voyager 2 in its flyby with Jupiter increased its
speed from 10 kms-1 to almost 30 kms-1 without
burning any fuel!
Variations in gravitational force
Gravitational force depends on the mass and is
inversely proportional to distance.
A body which has twice the mass of another will
have twice the gravitational force at the same
distance
Doubling the distance from a mass will reduce its
gravitational force to a quarter
The value of g on a planet's surface with the same
radius as Earth's but ½ the mass will be 9.8 / 2 =
4.9 ms-2
The value of g on a planet's surface with the same
mass as Earth's but ½ the radius will be 4 x 9.8 =
39.2 ms-2
On a local level, because g depends on mass, the
denser rocks will exert a higher gravitational
force.
Satellites can measure the surface variation of g
for the Earth and have produced colourful maps.
They show that where the crust is thinnest, the
ocean floor, g is smallest.
Revision questions - 4
1) Define Newton’s Law of Universal Gravitation.
2) Discuss the importance of Newton’s Law of Universal Gravitation in
understanding and calculating the motion of satellites.
3) Identify that a slingshot effect can be provided by planets for space
probes.
4) Discuss the factors that affect the strength of the gravitational force.
5) The International Space Station presently has a total mass of
approximately 227,000 kg and is in a LEO at an average altitude of
350 km. Calculate:
i. The value of g at this altitude
ii. The orbital velocity of the ISS
iii. Its period of rotation
iv. The work done to place it into orbit
v. Its kinetic energy
vi. The r3 / T2 ratio for the ISS and hence the period of rotation of the
Moon. (Moon – Earth distance = 3 x 10 5 km)
6) Describe a gravitational field in the region surrounding a massive
object in terms of its effects on other masses in it
4. “c”
To aether or not to aether?
In 1801 Thomas Young performed the double-slit experiment and
proved the wave nature of light. All waves known at that time
needed a medium in which to propagate, e.g. sound cannot travel
in a vacuum. But, no medium could be found for the propagation
of light and light was known to travel in a vacuum.
The luminiferous aether or ether was hypothesized as being the
medium that pervaded the entire universe (even solid objects) and
that it was the medium through which all electromagnetic waves
propagated.
The aether was given a set of properties:
The aether should: fill all of space and be stationary in space;
be perfectly transparent; permeate all matter; have a low
density and have great elasticity, it must be very light but
almost incompressible to allow light to travel so fast.
Yet, it must allow solid bodies to pass through it freely,
without any resistance, or the planets would be slowing down.
It is the relative motion of the Earth through the ether that
Michelson and Morley tried to measure.
M&MDetecting the
Aether Wind
The thought behind the Michelson-Morley (M&M)
experiment is better understood by considering
these two sketches of two swimmers (a similar
drawing was included in their original paper)
Swimmers A and B both swim at exactly the same
speed of 5 m/s. They both start and return at point
X.
In sketch 1 (no current)- both start and return at
the same time.
In sketch 2 (a 3 m/s current)
Swimmer A takes 12.5s to swim to Y (@ 8 m/s)
and 50s to swim back to X (@ 2 m/s). A total of
62.5s
Swimmer B has to take an angled course to point
P (because of the current) in order to return to
the same point. The course is longer (125 m) and
it takes him 25s for both trips a total of 50s
The river represents the aether; the two swimmers
represent Earth moving along and across the
aether. The movement of the solar system through
the aether is the current.
M&M designed equipment using an interferometer
that was calculated to be up to 40 times more
accurate than was needed.
M & M 's experiment
A null result, conclusion: NO AETHER ??
The paths A and B are
represented by the
swimmers A and B in the
preceding example. As
the Experiment is rotated
their respective arrival
times at the
interferometer screen
should change in the
presence of an aether
current. This change will
be easily detected by a
shift in the interference
fringes produced.
Even though the
experiment was repeated
many times, at different
times of the year
(position in Earth's orbit)
and at different altitudes.
NO SIGNIFICANT
CHANGE WAS EVER
DETECTED!
M&M' s null result
Albert Michelson and Edward Morley were experienced and rigorous
scientists (Michelson had just succeeded in making the most accurate
measurement of the speed of light of the time – a figure that is accurate to
this day).
Their experiment was well planned, well constructed and provided
reproducible results. They were not satisfied in repeating it a few times but
actually repeated it many times in different locations and at different times
during the year.
Both were firm believers in the aether theory, so if there was any bias at
all, it would have been for a positive result, not a null one. Both were very
reluctant in reaching the only conclusion possible from their results: that
the existence of the aether was severely in doubt. Both received the Nobel
prize for this fundamental “ null” experiment.
Many other physicists over the years (to this day!) have had trouble in
letting the aether theory go and many modifications of the aether theory
have been tested, but to date each test has failed.
Some years after the Michelson-Morley experiments, Albert Einstein
proposed his theory of special relativity. Even though Einstein did not set
out to prove that the aether did not exist, a result of his theory was that the
aether is not needed.
In contrast to the aether theory, Einstein's special theory of relativity has
had confirmation from many of experiments.
Inertial
frames of
reference.
An inertial frame of reference is one that is at
rest or that is moving at constant velocity
A non-inertial frame of reference is one that
is accelerating.
By definition all inertial frames of reference
are identical – all will produce the same
results to experiments carried out in them
Also by definition any mechanical
experiment carried out in an inertial frame of
reference will provide NO information
regarding the state of uniform motion or rest
of the frame.
i.e. You cannot tell whether you are moving
or not unless you look at another inertial
frame of reference and then you can measure
your relative speeds.
This is known as Galilean Relativity. Galileo
was the first to recognize that any
observation within the inertial frame of
reference would not help to find out whether
the frame of reference was at rest or moving
at a steady velocity.
Einstein took Galilean Relativity a step
forward, by including electromagnetism. He
proposed that any electromagnetic
experiment would also give the same results
in any inertial frame of reference.
Inertial or non-inertial?

A ballistic cart is good piece of equipment to demonstrate the difference
between the two frames of reference:
 In the first two experiments the ball propelled upward by the vertical
“cannon” returns cleanly to the cannon. An imaginary observer on the cart
cannot tell by just looking at the ball whether he is in 1 or 2 (assuming he
cannot see outside the train)

In the third experiment the cart is accelerating (pulled by a string) and the
ball is “left behind” the imaginary observer knows he is accelerating.
Einstein's Theory of Special Relativity
In 1864 James Maxwell derived a set equations that described the properties of electric and
magnetic fields and how they interact and give rise to each other. Using his equations he
was able to calculate the speed of light as being 3×108 ms-1 (c ) purely as a mathematical
consequence of the interaction of the two fields in a vacuum.
In 1905 Albert Einstein proposed that the laws of inertial frames must also apply to
Maxwell's equations just as they applied to the rest of physical laws.
Einstein's Special Relativity states that: “the Laws of Physics are the same in any inertial
frame and that includes any measurement of the speed of light - in any inertial frame the
speed of light will always be c (A consequence of special relativity is that there is no
natural rest-frames in the universe. Hence, not only is the aether not necessary, but also it
cannot exist as it would provide a natural frame at rest.)
The speed of light will be constant for all observers in any inertial frame of reference.
Einstein formulated his theory of relativity using
thought experiments. He did this to investigate
situations, that at the time, could not be tested
in reality.
Most of Einstein's deductions using his thought
experiments have since received experimental
confirmation. His thought experiments were
based on a deep understanding of physics and
on his natural genius.
His famous thought experiment that led him to
conclude that the aether model did not work for
light is simplified and illustrated here ----->
The result predicted by the aether model is that
he will not see his reflection. Since light can
only travel at C through the aether. The light
from the man's face cannot catch up to the
mirror to return as a reflection.
This violates the principle of inertial frames (in
an inertial frame of reference you cannot
perform any experiment to tell that you are
moving). Therefore, he must be able to see his
reflection
Einstein concluded that: the reflection will be
seen as normal so that the principle of inertial
frames would not be violated; he described the
aether model as superfluous.
Einstein's
thought
experiments
A man is sitting in a
train facing forwards.
The train is moving at
the speed of light. He
has a mirror in front of
him; will he be able to
see his reflection in
the mirror?
Spacetime
Einstein's prediction that the speed of light is a constant to all
observers (at rest or moving uniformly) led to a number of logical but
startling consequences.
It changed how we view time and space. The speed of light was
constant but space, time and even mass were relative to the observer.
The term “spacetime” was coined with this new understanding
Spacetime is viewed as a consequence of Einstein's 1905 Theory of
Special Relativity, but is treated more fully in Einstein's next cornerstone of modern physics: the Theory of General Relativity.
Special relativity gave birth to the concepts of time dilation, length
contraction, mass dilation and mass/energy equivalence. (it was only
one, of four paradigm-changing papers Einstein published in 1905 at
the age of 26!)
For many years there was no evidence for Einstein's Theory of Special
Relativity. However, starting with the confirmation of his prediction that
the light from a distant star would bend when passing close to the Sun,
evidence supporting the theory was not long in coming.
Relativity of simultaneity
Adapted from Einstein's thought experiment:
A train of length L has a door at both ends. They
are operated by means of a photocell.
When the light from a source in the middle of
the train hits the photocells the doors open
An observer A standing in the middle of the
train can turn the light on. When the train
reaches a uniform speed 'v' and when he
passes a station he turns the light on.
An other observer B is standing stationary at
the station as the train passes.
For A, the doors open at the same time.
He is in an inertial frame and light must reach
the photocells at the same time. (0.5L / C)
Observer B is in another inertial frame, he
sees the train passing and the light turn on .
But the light going to the front door takes a
little longer to reach the photocell because
the front of the train has moved a distance 'vt'
forward. i.e. (0.5L + vt) / C
Vice versa, the light going to the back of the
train takes a little less time. i.e.(0.5L - vt) / C
For B, the doors open at different times.
Note: The value of C is the same for both
observers.
Simultaneity is relative to the observer
Time dilation
Adapted from Einstein's thought experiment;
Each will see their own clock as

For this experiment we will use what can be
named Einstein's light clock:






A flash of light bounces back and forth between
two mirrors, d metres apart. Each time it hits the
top mirror the clock advances one click.
Time between clicks = 2d / C
Observer A is in a train moving at a constant
velocity 'v' past a railway station. A stationary
observer B is at the station.
Both observers have identical light clocks. In
each case they see their own light clock as
stationary (with respect to themselves) and the
other person's clock as a moving clock at
velocity v

But each will see the other person's clock as:
Using Pythagoras and substitution, it can be
shown that :
tv 
t0
1
v2
c2
Experimental evidence for time dilation
One of the implications of time dilation is that clocks moving
at relativistic speeds (speeds near the speed of light) will
appear to run significantly more slowly to an observer in
another frame of reference.
This was confirmed in 1963 (40 years after Einstein proposed
it) when the unstable subatomic particles called Muons were
measured to have longer half-lives at near light speeds. The
increase was exactly as predicted by the theory.
As technology has improved, so has our ability to measure
fractions of seconds more and more accurately. This
increased accuracy has allowed the theory to be confirmed
by comparing identical clocks at rest with ones in fast
moving jets. They too confirmed Einstein's formula.
The demand for more and more accurate GPS' has resulted in
having to correct, using the time dilation formula, for time
dilation effects due to the orbital speeds of the satellites in
the system.
Astronomical data from receding supernovae agrees with
time dilation calculations.
Length contraction
The length of any object in a moving frame will appear foreshortened in the
direction of motion, or contracted. The length is maximum in the frame in
which the object is at rest.
The length of an object measured within its inertial frame is called its proper
length Lo. The length of an object in a frame on relative motion to the
observer's frame is Lv
Length contraction (also known as Fitzgerald Contraction), like time dilation is
symmetrical and is relative to the observer.
lv  l0 1
v
2
c
2
Evidence for Length contraction
Length contraction and time dilation
are related as both are connected to
speed. This is best illustrated with the
Muon evidence for time dilation
The increase in the Muon's half-life is
seen from the perspective of an
observer on Earth as an example of
time dilation.
But from the Muon's point of view its
half-life is a constant and is not
changed by its speed. What has
changed is the distance outside its
frame of reference that it must cover
in that half-life.
The Muon "sees” that distance
foreshortened by length contraction
giving it time to cover it in its short life
span.
Atomic clocks and the metre.
In 1793 the French defined the unit of length to be
the metre as 1 ten millionth of the Earth's quadrant
passing through Paris.
For many years this standard was used as the
reference metre – the distance between two fine
lines on a platinum-iridium bar kept at a standard
temperature
Today the metre is defined using the speed of light
and using very small fractions of a second
measured by atomic clocks:
“The meter is the length of the path traveled by
light in a vacuum during a time interval of 1/C
seconds. Where C is the speed of light, taken as:
C = 299,792,458 ms-1
“A second is defined as the duration of
9,192,631,770 cycles of microwave light absorbed
or emitted by the hyperfine transition of cesium-133
atoms in their ground state undisturbed by external
fields.”
Mass – Energy equivalence
The mass–energy equivalence is the concept that mass is energy,
and that energy is mass.
In special relativity this relationship is expressed using the mass–
energy equivalence formula
E = mC2
(Where: E = energy; m = mass and C = the speed of light in a vacuum)
The formula (voted the best known formula of all time) was derived
by Albert Einstein, in 1905 in the paper "Does the inertia of a body
depend upon its energy-content?"
The concept of mass–energy equivalence unites the concepts of
conservation of mass and conservation of energy.
The mass–energy equivalence formula was used in the development
of nuclear reactors and of the atomic bomb. During Fission (or
Fusion) part of the mass is converted to energy. 1 kg of mass is
equivalent to 9 x 1016 Joules!
The Sun converts 4.26 million metric tons of mass to energy every
second
Mass-energy equivalence led Einstein to the concept of mass
dilation
Mass dilation
The term mass in special relativity usually refers to the rest-mass of
the object, which is the Newtonian mass as measured by an observer
with the same velocity as the object. The invariant mass is another
name for the rest-mass of single particles.
The term relativistic mass includes a contribution from the kinetic
energy of the body, and increases as the object's velocity increases..
(Where: mv= relativistic mass; m0 = rest mass; v is the velocity
relative to the observer and c = speed of light)
Mass dilation means that a particle moving at relativistic speeds will
have a greater mass than its rest mass when measured by an
observer in a different frame of reference.
Evidence for Mass dilation
Since the birth of nuclear energy the
evidence for the mass-equivalence concept
and its derivative the mass dilation equation
has been substantial.
Particle accelerators daily accelerate
subatomic nuclei to relativistic speed and
measure their masses using the mass
dilation relationship to achieve required
momenta for their collision experiments
Nuclear reactors produce energy by (mostly)
uranium fission, the energy out put can be
calculated using the mass-energy
equivalence formula
Both formulae are important in astronomy
when considering stellar evolution and
receding star systems
Every particle in the universe is in motion
and hence has a relativistic mass.
Deep-Space travel
A light-year (ly) is a measure of distance. It is the distance traveled by light
in one year in vacuum at the speed of 3 x 10 8 m. It is a very long distance (
9.4605284 × 1015 meters)
The closest star to the solar system is Proxima Centauri, it is 4.2 ly away.
Even traveling near light speed a space ship to Proxima Centauri would
have sustain its crew for a long voyage.
Special relativity would help in some ways and hinder us in others. Moving
at say 0.75C would shorten our path considerably because of length
contraction.
And we would return to Earth in less ship years than had passed on Earth,
because of time dilation.
But mass dilation would make it almost impossible for us to reach
relativistic speeds as it would require increasing and vast amounts of
energy to accelerate to those speeds.
Our Galaxy, just one of billions in space, it is 100,000 ly across. IntraGalactic travel will need the discovery of other means of travel:
hyperspace, wormholes or something that has not even been thought of.
NASA projects that we may be able to get to Mars by 2030. Mars is “just”
11 light-minutes away.
Revision questions - 5
1) Outline the features of the aether model.
2) Describe and evaluate the Michelson-Morley experiment.
3) Discuss the role of the Michelson-Morley experiments in
making determinations about competing theories
4) Interpret the results of the Michelson-Morley experiment
5) Outline the nature of inertial frames of reference
6) Describe an investigation you performed to distinguish
between non-inertial and inertial frames of reference
7) Discuss the principle of relativity
8) Describe the significance of Einstein’s assumption of the
constancy of the speed of light
9) Identify the implications of C being a constant to all observers.
Revision questions - 6
1) Explain qualitatively and quantitatively the consequence of the
relativity of simultaneity.
2) Explain qualitatively and quantitatively the consequence of the
equivalence between mass and energy.
3) Explain qualitatively and quantitatively the consequence of length
contraction.
4) Explain qualitatively and quantitatively the consequence of mass
dilation.
5) Explain qualitatively and quantitatively the consequence of time
dilation.
6) Discuss the implications of mass increase, time dilation and length
contraction for space travel.
7) Discuss the relationship between theory and the evidence supporting
it, using Einstein’s predictions based on relativity that were made
many years before evidence was available to support it.
8) Discuss the concept that length standards are defined in terms of
time in contrast to the original metre standard.
9) Analyse and interpret some of Einstein’s thought experiments
involving mirrors and trains and discuss the relationship between
thought and reality.