Transcript time vs velocity

One just isn’t enough!
Four Major Types of
Two Dimensional Motion
1. Projectile Motion
2. Circular Motion
3. Rotational Motion
4. Periodic Motion
Projectile motion problems are best solved by
treating horizontal and vertical motion separately
because perpendicular vectors are independent
*IMPORTANT*
Gravity only affects vertical motion.
There are two general types
of projectile motion situations.
1. object launched horizontally
2. object launched at an angle
The path of a projectile is called the trajectory
Object Launched Horizontally
vx = initial horizontal velocity
h = initial height
above ground
t = total time in the air Rx = horizontal range
IMPORTANT FACTS
There is no horizontal acceleration.
There is no initial vertical velocity.
The horizontal velocity is constant.
Time is the same for both vertical and horizontal.
horizontal
Rx = vxt
vertical
h = 0.5gt2
Horizontal projectile examples
Object Launched at an Angle
v = initial velocity
q = launch angle
h = maximum height
t = total time in air
Rx = horizontal range
IMPORTANT FACTS
The horizontal velocity is constant.
It rises and falls in equal time intervals.
It reaches maximum height in half the total time.
Gravity only effects the vertical motion.
horizontal
vx = v cosq
Rx = vxt
vertical
vy = v sinq
h = vyt/4
t = -2vy/g
Angled projectile examples:
Learn more about projectile motion
at these links:
link1, link2, link3, link4, link5, link6
View projectile motion simulations at:
link1, link2, link3, link4, link5, link6
Suggested Constructivist Activities
Students use simulations to complete data tables
and make graphs of the following situations:
• constant initial velocity versus how the horizontal
range changes with angle; plot “range vs angle”
• constant initial velocity versus how total time in air
changes with angle; plot “total time vs angle”
• constant initial velocity versus how maximum height
changes with angle; plot “height vs angle”
• constant angle versus how the horizontal range
changes with initial velocity; plot “range vs velocity”
• constant angle versus how the total time in the air
changes with initial velocity; plot “time vs velocity”
• constant angle versus how the maximum height
changes with initial velocity; plot “height vs velocity”
object moves in
circular path
about an External point
(“orbits”)
According to Newton’s First Law of Motion,
objects move in a straight line unless a force
makes them turn. An external force is necessary
to make an object follow a circular path.
This force is called a
CENTRIPETAL (“center seeking”) FORCE.
Since every unbalanced force causes an object
to accelerate in the direction of that force
(Newton’s Second Law), a centripetal force
causes a CENTRIPETAL ACCELERATION. This
acceleration results from a change in direction,
and does not imply a change in speed,
although speed may also change.
Centripetal force and acceleration may be caused by:
•gravity - planets orbiting the sun
•friction - car rounding a curve
•a rope or cord - swinging a mass on a string
In all cases, a mass m moves in a circular path
of radius r with a linear speed v. The time to make
one complete revolution is known as the period, T.
The speed v is the
v circumference divided by the period.
r
m
v = 2pr/T
If the force that causes it to turn is removed, it
will travel in a straight line tangent to the circular
path at the point the force was removed.
The formula for centripetal acceleration is:
ac =
2
v /r
and centripetal force is:
Fc = mac =
2
mv /r
m = mass in kg
v = linear velocity in m/s
Fc = centripetal force in N
r = radius of curvature in m
ac = centripetal acceleration in m/s2
For vertical circles: V is not constant through the
circular path
-V accelerates downward
-V decelerates upward
-Fc is minimum on top and
maximum on bottom
because Fc = mV2
R
-At the top, the force in the
string and force of weight both
act toward the center (downward)
-At the bottom, the force in the
string acts toward the center and
the weight acts downward in an
opposite direction.
FW
FT
At the top where V = Vmin :
Fc is equal to FT and FW
(FT + FW) since both act
toward the center.
Therefore:
FC(top) = FT + FW = mVmin2
r
At the bottom where V = Vmax,
Fc is FT minus Fw (since they now
act in opposite directions).
Therefore,
Fc(bottom) = FT – Fw = mVmax2
r
Since Vmax is at the bottom, then Fc
is greater at the bottom; therefore
FT is greatest at the bottom and
greater than the Fc at the bottom
by an amount equal to the Fw.
You can calculate the critical velocity, or minimum
velocity required to maintain a circular path (such as
that of a satellite about the earth or a roller coaster at
the top of a loop) with Vmin = rg
from: mVmin2 = Fw (since FT = 0 at minimum
r
velocity b/f falling out
of circular path)
mVmin2 = mg
r
rmg = mVmin2
rg = Vmin2
rg = Vmin
Learn more about circular motion
at these links:
link1, link2, link3, link4, link5
View circular motion simulations at:
link1, link2, link3, link4
object moves in
circular path about
an Internal point
or axis
(“rotates” or “spins”)
The amount that an object rotates is its
angular displacement.
angular displacement, q, is given in
degrees, radians, or rotations.
1 rotation = 360 deg = 2p radians
The time rate change of an object’s
angular displacement is its
angular velocity.
angular velocity, w, is given in
deg/s, rad/s, rpm, etc...
Radians- an angle of one radian
is the angle that, when placed
with vertex at the center of a
circle, substends on the
circumference an arc equal in
length to the radius of the
circle: when s = r, q = 1 radian
Since C = 2pr, 360o= 2p radians and 1 radian = 57.3o
(a pure number; no unit)
Angular velocity
is a vector.
The “right-hand rule” describes
the direction of angular velocity.
The direction of w is the
direction of the thumb of the
right hand when the fingers curl
in the rotational direction.
The time rate change of an object’s
angular velocity is its
angular acceleration.
Angular acceleration, a, is given in
deg/s2, rad/s2, rpm/s, etc...
Formulas for rotational motion follow an
exact parallel with linear motion formulas.
The only difference is a change in variables
and a slight change in their meanings.
Constant
LINEAR
ROTATIONAL
vf = vi + at
d = vavt
vav = (vf + vi)/2
d = vit + 0.5at2
wf = wi + at
q = wavt
vav = (wf + wi)/2
vf2 = vi2 + 2ad
wf2 = wi2 + 2aq
q = wit +
2
0.5at
Forces that stop, start, or change the direction of rotation
are called torque. Torque is equal to the product of the force
applied and the lever arm
The lever arm ( d, below) is the perpendicular distance from
the axis of rotation to a line along which the force acts (the
“F”, below. It was extended with dashed line downward to
make it long enough to draw in the lever arm perpendicularly)
Rotational Inertia (I) is the resistance of a rotating object
to changes in its angular velocity. We can convert the
equation F = ma into one to be used for rotary motion: T = aI
(Where T = torque)
Rotational inertia depends not only on the mass of the
rotating object but also on the distribution of the mass:
any motion in which
the path of the object
repeats itself in equal
time intervals
MOTION
The simple pendulum
is a great example of
this type of motion.
When an object in periodic motion has one position in the
motion where it is in equilibrium and is subject to a restoring
force at all other positions that varies linearly with
displacement from the equilibrium position, the object is in
simple harmonic motion. Examples include a pendulum, an
object moving up and down on a spring, and a guitar string
vibrating.
•the restoring force, so called because it restores the
object to the equilibrium position, may be gravity or the
inherent stretch of a spring or guitar string.
•The period (T) is the time required to complete one full
cycle of motion.
* The amplitude is the maximum distance the object moves
from the equilibrium position.
The swing of a pendulum is
simple harmonic motion:
* the length of the string is
l and the force of the
suspended object is the
weight which is resolved
into 2 components.
•The Fll along the direction
of the string and the
•Fl is at right angles to the
direction of the string and
always directed toward the
equilibrium position.
•The Fl is the restoring
force for a pendulum.
The period, T, of a simple pendulum
(time needed for one complete cycle)
is approximated by the equation:
l
T  2p
g
where l is the length of the pendulum
and g is the acceleration of gravity.
The frequency of a pendulum is the number of
complete cycles in one second. It is found by f =
1/T. The reciprocal of the period.
For pendulums, the period is independent of mass or material of
the pendulum (neglecting air resistance), independent of
amplitude (if the arc is small), and inversely proportional to the
square root of the acceleration of gravity.
Learn more about pendulums and
periodic motion at these links:
link1, link2, link3, link4, link5
View pendulum simulations at:
link1, link2, link3, link4, link5