Projectile Motion
AP Physics: Mechanics
2D Motion
Monkey and the Hunter
Can a monkey be hit
with a tranquilizer dart
from far away if the
monkey drops from a
branch at the same
time as the dart is
shot?
Two Assumptions
 The rate of gravitational acceleration is constant
 Air resistance is negligible
 What shape trajectory do we expect?
Hang-Time
QUESTION:
How can the “hang-time” or “time of flight” of a horizontally fired
projectile be extended? What factors affect the flight time of a
projectile?
CHALLENGE:
1) Design and carry out a quick experiment using a tennis ball, a
stopwatch, and a tape measure that will answer the questions
above.
2) Put the data that you collect on a whiteboard and convince the
class that you are correct by the end of the period.
3) Turn in one sheet per group that has your data and explanations.
Hang-time
DOES NOT depend upon initial horizontal
velocity
No forces in the x direction,
therefore no change in velocity in the x
direction.
The ball stops due to gravity, so only a
higher launch point will increase the hangtime of a horizontally fired projectile.
Hang-Time
v y  v 0 y  at
v y  gt
v0x
ay=g

y
y 
1
2
y 
x
v x  v 0 x  at
v x  v 0x

t

2
a yt  v 0yt
1
2
gt
2
2 y
g
Horizontal Range
 QUESTION:
What determines the distance that a horizontally fired projectile will
travel?
 CHALLENGE:
1) Predict the Range of a horizontally fired projectile. Check your
prediction and calculate your percent error.
2) Carry out this experiment using a steel ball and the provided
trajectory apparatus.
3) For this experiment you should turn in the work that shows your
prediction. Explain the reasoning behind your prediction. Show
why you believe that you had a high or low percent error.
4) You should also include a general equation for the range of a
horizontally fired projectile.
Horizontal Range
th 
v0x
d
2h
g
v x  v0 x
ay=g
R  v0 x t h
h
v0 x 
d
t
R
R 
d
2h
t
g
Rocket Science
 QUESTION:
How does angle affect the time of flight and the range of a projectile?
 CHALLENGE:
1) Answer the question above. Provide evidence (data tables and graphs) to
support your findings. Consider and report sources of error and ways to
improve the lab.
2) Make predictions about which angle will make the greatest and least range.
Find the initial velocity of the rocket and predict its range. Compare this to the
actual range.
3) Make predictions about which angle will make the rocket spend the most
time in the air. Compare your calculated hang-times to actual values from a
stopwatch.
4) Turn one response in per group.
Rocket Science
 SAFETY: The rockets are potentially VERY dangerous. The
rockets leave the launcher at speeds exceeding 60 mph. If you do
not give complete attention to the lab, you or someone else could
be seriously injured.
 NEVER look directly down at the rocket while it is on the launcher.
 ALWAYS disconnect the air supply before touching a rocket which is on
the launcher.
 NEVER make any attempts whatsoever that even remotely look to me
like you are going to launch the rocket at another individual.
 NEVER attempt to catch a rocket while in flight.
 NEVER launch the cap by itself.
 NOTE:
Since this lab is being done on the football field, we
will measure everything in yards instead of meters. Make sure to
convert your value of acceleration due to gravity (g).
(R/2,h) & (R,0)
What are R horizontal range and h
maximum height in terms of v0, g,
and θ0?
v y  v 0 sin  0   gt
vy  0
0  v 0 sin  0   gt
t top 
v 0 sin
g
0 
t top 
v 0 sin
0 
y  v 0 sin  0 t 
g
y h
h  v 0 sin  0 

h
v 0 sin
g

v 0 sin  0 
0
g
2

2
gt
2
substitute ttop
2
1 v 0 sin  0  
 g 

2 
g

1 v 0 sin  0 
2
1
g
2
h
1 v 0 sin  0 
2
g
2
(R/2,h) & (R,0)
What are R horizontal range and h
maximum height in terms of v0, g,
and θ0?
t top 
v 0 sin
0 
t R  2t top 
g
xR
R  v 0 cos  0 t

2 v 0 sin  0 
g
t R  2t top 
2 v 0 sin  0 
g
R  v 0 cos  0 t
Substitute 2ttop
 
2 v 0 sin  0  
R  v 0 cos  0 

g


R 

2 v cos  0 sin  0 
2
0
g
R 
2
0
v sin 2 0
g
How can we express
maximum range?
R 
2
0
v sin 2 0
g
if  0  45 
sin 2  0  sin 90   1
When is this
expression a
maximum?
R max 
v
2
0
g
Kinematic Equations in 2D
v x  v 0 x  v 0 cos  0  cons tan t
2
x  v 0 x t  v 0 cos  0 t
v
ax  0
y 
  tan  
x 
1
v y  v 0 y  gt  v 0 sin  0   gt
y  v 0yt 
vx  vy
1
2
gt
2
 v 0 sin  0 t 
ay  g

1
2
gt
2
r  v 0t 
1
2
gt
2
2
The shape of a projectile’s
trajectory
x  v 0 cos  0 t
y  v 0 sin  0 t 
t
x
v 0 cos  0 
Substitute

2

1 
x
y  v 0 sin  0 
 g 

v 0 cos  0 2 v 0 cos  0 
x
1
2
gt
2
The shape of a projectile’s
trajectory
2

1 
x
y  v 0 sin  0 
 g 

v 0 cos  0 2 v 0 cos  0 
x


g
2
y  tan  0 x  
x
2
2
2 v 0 cos  0 
Quadratic!!!

Parabola!!!
Conclusions
Projectile motion is a superposition of
two motions.
1) Constant velocity motion in the initial
direction.
2) The motion of a particle freely falling
in the vertical direction under constant
acceleration.