v o =v o cos 
x
h
v o =v o sin 
vo
y

xT
A monkey hangs from a tree. I plan to shoot him with a rifle. I know that
the very instant I pull the trigger, the monkey will get scared and let go
of the branch. The monkey will be in free fall, while the bullet will
exhibit projectile motion. Ignore all forces, such as air resistance (drag)
and buoyancy. Consider only the force of gravity to be relevant to this
experiment
1
v o =v o cos 
x
h
v o =v o sin 
vo
y

xT
At what angle should I point
the gun?
2
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
x T =Horizontal Distance to the Target.
y T =Vertical position of the Target.
3
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
The monkey is in Freefall. The
monkey's initial height is h. The
equations to the right show the
monkey's position as a function of
time. The initial velocity of the
monkey is zero since the monkey
starts from rest.
Monkey ( Freefall)
1
y T = y o v o t− g t 2
2
1
y T = y o v o t− g t 2
2
1
y T =h− g t 2
2
T
y
T
y
4
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
The bullet follows the trajectory of
a projectile. The bullet's y position
is found by:
1 2
y T = y o v o t− g t
2
1 2
y T =v o t− g t
2
y
y
y
Monkey ( Freefall)
1
y T = y o v o t− g t 2
2
1
y T = y o v o t− g t 2
2
1
y T =h− g t 2
2
T
y
T
y
5
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
Bullet (Projectile Motion)
1 2
y T = y o v o t− g t
2
1 2
y T =v o t− g t
2
y
y
y
Monkey ( Freefall)
1
y T = y o v o t− g t 2
2
1
y T = y o v o t− g t 2
2
1
y T =h− g t 2
2
T
y
T
y
6
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
Bullet (Projectile Motion)
1 2
y T =v o t− g t
2
y
Monkey ( Freefall)
1
y T =h− g t 2
2
These two heights have to be equal if the bullet is to hit
the monkey.
1
1
y T =h− g t 2 =v o t − g t 2
2
2
1
1
h− g t 2 =v o sin t − g t 2
2
2
y
7
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
Bullet (Projectile Motion)
1 2
y T =v o t− g t
2
y
Monkey ( Freefall)
1
y T =h− g t 2
2
These two heights have to be equal if the bullet is to hit the monkey.
1
1
y T =h− g t 2 =v o t − g t 2
2
2
1
1
h− g t 2 =v o sin t − g t 2
2
2
h=v o sin  t
y
8
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
Bullet (Projectile Motion)
1 2
y T =v o t− g t
2
y
1
1
y T =h− g t 2 =v o t − g t 2
2
2
1
1
h− g t 2 =v o sin t − g t 2
2
2
h=v o sin  t
y
Monkey ( Freefall)
1
y T =h− g t 2
2
The time can be calculated from
the constant x component of the
velocity.
x T =v o t
xT
t=
vo
x
x
9
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
1
1
y T =h− g t 2 =v o t − g t 2
2
2
1
1
h− g t 2 =v o sin t − g t 2
2
2
h=v o sin  t
y
x T =v o t
xT
xT
t= =
v o v o cos 
x
x
10
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT
1
1
y T =h− g t 2 =v o t − g t 2
2
2
1
1
h− g t 2 =v o sin t − g t 2
2
2
xT
h=v o sin 
v o cos 
y

x T =v o t
xT
xT
t= =
v o v o cos 
x
x

11
At what angle should I point
the gun?
v o =v o cos 
x
h
v o =v o sin 
vo
y
yT

xT

xT
h=v o sin 
v o cos 

 
xT
h=sin 
cos 
h
=tan 
xT
12
At what angle should I point
the gun?
v o =v o cos 
x
v o =v o sin 
y
h
vo
=tan
−1

h
xT
yT
xT

xT
h=v o sin 
v o cos 
 
xT
h=sin 
cos 
h
=tan 
xT
h
=tan−1
xT
 

h

xT
Point the gun at
the monkey's
initial position.
13
At what angle should I point
the gun?
v o =v o cos 
x
v o =v o sin 
y
h
vo
=tan
−1
yT

h
xT
xT
=tan
−1
 
h
xT
1 2
y T =v o t − g t
2
xT
t=
v o cos 
y
The faster the initial velocity, the
higher the yT will be.

2
  
 
xT
xT
1
y T =v o sin 
− g
v o cos 
2
v o cos 
xT
1
y T = x T tan − g
2
v o cos 
2
14