Transcript Vectors and Projectiles

Vectors and Projectiles
• A vector is a quantity which has both magnitude
and direction. Examples of vectors include
displacement, velocity, acceleration, and force. To
fully describe one of these vector quantities, it is
necessary to tell both the magnitude and the
direction. For instance, if the velocity of an object
were said to be 25 m/s, then the description of
the object's velocity is incomplete; the object
could be moving 25 m/s south, or 25 m/s north
or 25 m/s southeast. To fully describe the object's
velocity, both magnitude (25 m/s) and direction
(e.g., south) must be stated.
Most of us are accustomed to the following form
of mathematics:
6 + 8 = 14.
Yet, we are extremely uneasy about this form of
mathematics:
• 6 + 8 = 10 and 6 + 8 = 2 and 6 + 8 = 5.
Vectors are quantities which include a
direction
• There are a number of methods for carrying out the
addition of two (or more) vectors. The most common
method is the head-to-tail method of vector addition. Using
such a method, the first vector is drawn to scale in the
appropriate direction.
• The second vector is then drawn such that its tail is
positioned at the head (vector arrow) of the first vector.
The sum of two such vectors is then represented by a third
vector which stretches from the tail of the first vector to
the head of the second vector. This third vector is known as
the resultant - it is the result of adding the two vectors.
• The resultant is the vector sum of the two individual
vectors. Of course, the actual magnitude and direction of
the resultant is dependent upon the direction which the
two individual vectors have.
• This principle of the head-totail addition of vectors is
illustrated in the animation
below. In each frame of the
animation, a vector with
magnitude of 6 (in green) is
added to a vector with
magnitude of 8 (in blue).
The resultant is depicted by
a black vector which
stretches from the tail of the
first vector (8 units) to the
head of the second vector (6
units).
• As can be seen from this, 8 + 6 could be equal to
14, but only if the two vectors are directed in the
same direction.
• All that can be said for certain is that 8 + 6 can
add up to a vector with a maximum magnitude of
14 and a minimum magnitude of 2.
• The maximum is obtained when the two vectors
are directed in the same direction. The minimum
s obtained when the two vectors are directed in
the opposite direction.
Vector Addition:
The Order Does NOT Matter
• The question often arises as to the importance
of the order in which the vectors are added.
For instance, if five vectors are added - let's
call them vectors A, B, C, D and E - then will a
different resultant be obtained if a different
order of addition is used. Will A + B + C + D + E
yield the same result as C + B + A + D + E or D
+ E + A + B + C?
• the order in which two or more vectors are
added does not effect the outcome. Adding A
+ B + C + D + E yields the same result as adding
C + B + A + D + E or D + E + A + B + C! The
resultant, shown as the green vector, has the
same magnitude and direction regardless of
the order in which the five individual vectors
are added.
The Plane and The Wind
• Each plane is heading south with a speed of 100 mi/hr. Each plane
flies amidst a wind which blows at 20 mi/hr. In the first case, the
plane encounters a tailwind (from behind) of 20 mi/hr. The
combined effect of the tailwind and the plane speed provide a
resultant velocity of 120 mi/hr. In the second case, the plane
encounters a headwind (from the front) of 20 mi/hr. The combined
effect of the headwind and the plane speed provide a resultant
velocity of 80 mi/hr. In the third case, the plane encounters a
crosswind (from the side) of 20 mi/hr. The combined effect of the
headwind and the plane speed provide a resultant velocity of 102
mi/hr (directed at an 11.3 degree angle east of south). These three
resultant velocities can be determined using simple rules of vector
addition. In the case of the crosswind, the Pythagorean Theorem
and SOH CAH TOA are utilized to determine the magnitude and the
direction of the resultant velocity.
The River Boat
• Assuming that in each case the motor of the
boat propels it across the river with the same
force, in which case (with or without a
current) will the boat make it across the shore
the soonest? You might be surprised by your
observation.
Parabolic Motion of Projectiles
• A projectile is an object upon which the only
force is gravity. Gravity, being a downward force,
causes a projectile to accelerate in the downward
direction. The force of gravity could never alter
the horizontal velocity of an object since
perpendicular components of motion are
independent of each other.
• A vertical force does not effect a horizontal
motion. The result of a vertical force acting upon
a horizontally moving object is to cause the
object to deviate from its otherwise linear path.
The Monkey and Zookeeper
• There is an interesting monkey down at the zoo. The
monkey spends most of its day hanging from a limb of
a tree.
• The zookeeper feeds the monkey by shooting bananas
from a banana cannon to the monkey in the tree.
• This particular monkey has a habit of dropping from
the tree the moment that the banana leaves the
muzzle of the cannon.
• The zookeeper is faced with the dilemma of where to
aim the banana cannon in order to hit the monkey.
• If the monkey lets go of the tree the moment that the
banana is fired, then where should she aim the banana
cannon?
• If there were no gravity, then what would
happen if the banana was shot at the
monkey?
• What path would the banana take and would
it hit the monkey?
Gravity-Free Environment
Aiming Above the Monkey
• If there was gravity acting upon both the
monkey and the banana (the usual situation),
then what would happen if the banana was
shot above the monkey? What paths would
the banana and the monkey take? Would the
banana fall (below the straight-line path) and
hit the monkey as the monkey drops from the
tree? Or would the banana miss the monkey,
passing over his head?
Aiming at the Monkey - Fast
• If there was gravity acting upon both the monkey
and the banana (the usual situation), then what
would happen if the banana was thrown at the
monkey with a fast speed? What paths would the
banana and the monkey take? Would the banana
fall (below the straight-line path) and hit the
monkey as the monkey drops from the tree? Or
would the banana merely move in a straight line
and hit the monkey immediately? Or would the
banana miss the monkey, passing over his head
(or even below his head)?
Aiming at the Monkey - Slow
• If there was gravity acting upon both the monkey
and the banana (the usual situation), then what
would happen if the banana was thrown at the
monkey with a slow speed? What paths would
the banana and the monkey take? Would the
banana fall (below the straight-line path) and hit
the monkey as the monkey drops from the tree?
Or would the banana not have enough speed to
hit the monkey as it was falling (presumably,
because the monkey would fall faster than
the slow banana)?
Horizontally Launched Projectiles
• Imagine a cannonball being launched from a
cannon atop a very high cliff. Imagine as well
that the cannonball does not encounter a
significant amount of air resistance. What will
be the path of the cannonball and how can
the motion of the cannonball be described?
Non-Horizontally Launched Projectiles
• Imagine a cannonball being launched at an angle
from a cannon atop a very high cliff. Imagine as
well that the cannonball does not encounter a
significant amount of air resistance. What would
be the path of the cannonball and how could the
motion of the cannonball be described? The
animation below depicts such a situation. The
path of the cannonball is shown; additionally, the
horizontal and vertical velocity components are
represented by arrows in the animation.
Maximum Range
• Imagine a cannonball launched from a cannon at
three different launch angles - 30-degrees, 45degrees, and 60-degrees. The launch speed is
held constant; only the angle is changed. Imagine
as well that the cannonballs do not encounter a
significant amount of air resistance. How will the
trajectories of the three cannonballs compare?
Which cannonball will have the greatest range?
Which cannonball will reach the highest peak
height before falling? Which cannonball will reach
the ground first?
The Plane and The Package
• Consider a plane moving with a constant
speed at an elevated height above the Earth's
surface. In the course of its flight, the plane
drops a package from its luggage
compartment. What will be the path of the
package and where will it be with respect to
the plane? And how can the motion of the
package be described? The animation below
depicts such a situation.
The Truck and The Ball
• Imagine a pickup truck moving with a constant
speed along a city street. In the course of its
motion, a ball is projected straight upwards by a
launcher located in the bed of the truck. Imagine
as well that the ball does not encounter a
significant amount of air resistance. What will be
the path of the ball and where will it be located
with respect to the pickup truck? How can the
motion of the ball be described? And where will
the ball land with respect to the truck?
Satellite Motion
• A satellite is often thought of as being a
projectile which is orbiting the Earth. But how
can a projectile orbit the Earth?
• Doesn't a projectile accelerate towards the
Earth under the influence of gravity?
• And as such, wouldn't any projectile
ultimately fall towards the Earth and collide
with the Earth, thus ceasing its orbit?
• These are all good questions and represent
stumbling blocks for many students of physics.
We will discuss each question here. First, an
orbiting satellite is a projectile in the sense that
the only force acting upon an orbiting satellite is
the force of gravity.
• Most Earth-orbiting satellites are orbiting at a
distance high above the Earth such that their
motion is unaffected by forces of air resistance.
Indeed, a satellite is a projectile.
• Second, a satellite is acted upon by the force of
gravity and this force does accelerate it towards
the Earth. In the absence of gravity a satellite
would move in a straight line path tangent to the
Earth. In the absence of any forces whatsoever,
an object in motion (such as a satellite) would
continue in motion with the same speed and in
the same direction. This is the law of inertia. The
force of gravity acts upon a high speed satellite to
deviate its trajectory from a straight-line inertial
path. Indeed, a satellite is accelerating towards
the Earth due to the force of gravity.
• Finally, a satellite does fall towards the Earth;
only it never falls into the Earth. To
understand this concept, we have to remind
ourselves of the fact that the Earth is round;
that is the Earth curves. In fact, scientists
know that on average, the Earth curves
approximately 5 meters downward for every
8000 meters along its horizon. I
• If you were to look out horizontally along the horizon of the Earth
for 8000 meters, you would observe that the Earth curves
downwards below this straight-line path a distance of 5 meters. In
order for a satellite to successfully orbit the Earth, it must travel a
horizontal distance of 8000 meters before falling a vertical distance
of 5 meters. A horizontally launched projectile falls a vertical
distance of 5 meters in its first second of motion. To avoid hitting
the Earth, an orbiting projectile must be launched with a horizontal
speed of 8000 m/s. When launched at this speed, the projectile will
fall towards the Earth with a trajectory which matches the
curvature of the Earth. As such, the projectile will fall around the
Earth, always accelerating towards it under the influence of gravity,
yet never colliding into it since the Earth is constantly curving at the
same rate. Such a projectile is an orbiting satellite.
• To further understanding the concept of a projectile
orbiting around the Earth, consider the following thought
experiment. Suppose that a very powerful cannon was
mounted on top of a very tall mountain. Suppose that the
mountain was so tall that any object set in motion from the
mountaintop would be unaffected by air drag. Suppose that
several cannonballs were fired from the cannon at various
speeds - say speeds of 8000 m/s, less than 8000 m/s, and
more than 8000 m/s. A cannonball launched with speeds
less than 8000 m/s would eventually fall to the Earth. A
cannonball launched with a speed of 8000 m/s would orbit
the Earth in a circular path. Finally, a cannonball launched
with a speed greater than 8000 m/s would orbit the Earth
in an elliptical path.
Launch Speed less than 8000
m/sProjectile falls to Earth
Launch Speed less than 8000
m/sProjectile falls to Earth
Launch Speed equal to 8000 m/sProjectile
orbits Earth - Circular Path
Launch Speed greater than 8000
m/sProjectile orbits Earth - Elliptical Path
• Two final notes should be made about
satellite motion. First, the 8000 m/s figure
used in the above discussion applies to
satellites launched from heights just above
Earth's surface. Since gravitational influences
decrease with the height above the Earth, the
orbital speed required for a circular orbit is
less than 8000 m/s at significantly greater
heights above Earth's surface.
• Second, there is an upper limit on the orbital
speed of a satellite. If launched with too great
of a speed, a projectile will escape Earth's
gravitational influences and continue in
motion without actually orbiting the Earth.
Such a projectile will continue in motion until
influenced by the gravitational influences of
other celestial bodies.