Projectile Motion

AP Physics: Mechanics2D 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 AssumptionsThe rate of gravitational acceleration is constant

Air resistance is negligible

What shape trajectory do we expect?

Hang-TimeQUESTION:

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-timeDOES 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 hang-time of a horizontally fired projectile.

Hang-Time

vy v0y at

v0x

y

x

ay=g

vy gt

y 1

2ayt

2 v0yt

y 1

2gt 2

t 2y

g

vx v0x at

vx v0x

Horizontal RangeQUESTION:

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

v0x

h

R

ay=g

2h

ht

g

0x xv v

0x hR v t

2d hR

t g

0x

dv

t

d

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 ScienceSAFETY: 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?

vy v0 sin0 gt

vy 0

0 v0 sin0 gt

t top v0 sin0

g

t top v0 sin0

g

y v0 sin0 t 1

2gt 2

y h substitute ttop

h v0 sin0 v0 sin0 g

1

2g

v0 sin0 g

2

h v0 sin0 2

g

1

2

v0 sin0 2

g

h 1

2

v0 sin0 2

g

(R/2,h) & (R,0)

What are R horizontal range and h maximum height in

terms of v0, g, and θ0?

t top v0 sin0

g

tR 2t top 2 v0 sin0

g

R v0 cos0 t

x R

R v0 cos0 t

tR 2t top 2 v0 sin0

g

Substitute 2ttop

R v0 cos0 2 v0 sin0 g

R 2 v0

2 cos0 sin0 g

R v0

2 sin20

g

How can we express maximum range?

R v0

2 sin20

g

When is this expression a maximum?

if 0 45sin20 sin90 1

Rmax v0

2

g

Kinematic Equations in 2D

v vx2 vy

2

tan 1 y

x

r v0t 1

2gt 2

vy v0y gt v0 sin0 gt

y v0yt 1

2gt 2 v0 sin0 t 1

2gt 2

ay g

vx v0x v0 cos0 constant

x v0xt v0 cos0 tax 0

The shape of a projectile’s trajectory

y v0 sin0 t 1

2gt 2

x v0 cos0 t

t x

v0 cos0

Substitute

y v0 sin0 x

v0 cos0

1

2g

x

v0 cos0

2

The shape of a projectile’s trajectory

y v0 sin0 x

v0 cos0

1

2g

x

v0 cos0

2

y tan0 x g

2v02 cos2 0

x2

Quadratic!!!

Parabola!!!

ConclusionsProjectile 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.