User's Manual
012-05479B Mini Launcher
31
®
Experiment 7: Varying Angle To Maximize Height on a Wall
EQUIPMENT NEEDED
– Mini Launcher and steel ball – Plumb bob
– Measuring tape or meter stick – Carbon paper
– White paper – Board to protect wall
Purpose
The purpose of this experiment is to find the launch angle which will maximize the height at which the
ball strikes a vertical wall for a ball launched at a fixed horizontal distance from the wall.
Theory
When the ball is launched at an angle from a fixed distance, x, from a vertical wall, it hits the wall at a
height y given by:
y
=
y
0
+
v
0
sin
q
t
-
1
2
gt
2
where y
0
is the initial height of the ball, v
0
is the initial speed of the ball as it leaves the muzzle, θ is the
angle of inclination above horizontal, g is the acceleration due to gravity, and t is the time of flight. The
range is the horizontal distance, x, between the muzzle of the Launcher and the place where the ball hits,
given by x = (v
0
cosθ)t. Solving for the time of flight from the equation for x gives
t =
x
v
0
cos
θ
Substituting for t in the equation for y gives
y
=
y
0
+
x
tan
q
-
gx
2
2
v
0
2
cos
2
q
To find the angle that gives the maximum height, y, set dy/dθ equal to zero and solve for the angle.
y
=
y
0
+
v
0
sin
q
t
-
1
2
gt
2
Solving for the angle gives
tan
θ
max
=
v
0
2
gx
Since the second derivative is negative for θ
max
,
the angle is a maximum.
To find the initial velocity of the ball, the fixed
distance x and the maximum height y
max
can be
used. Solve the y-equation for v
0
and plug in the
values for y
max
, θ
max
, and x.
Setup
➀ Clamp the Mini Launcher near one end of a
sturdy table with the Launcher facing the wall at
a distance of about 1 meter from the wall.
➁ Put a vertical board up to protect the wall.
➂ Test fire the ball (on the long range setting) a
few times to find approximately at which angle
the ball strikes the wall at a maximum height.
q
y
0
x
y
v
0
9
0
8
0
7
0
60
5
0
4
0
30
2
0
1
0
0
1
0
2
0
3
0
40
M
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M
E-6825
CAUTION!
W
E
A
R
S
A
F
E
T
Y
G
L
A
S
S
E
S
W
H
E
N
I
N
U
S
E
.
D
O
N
'
T
P
U
S
H
P
I
S
T
O
N
W
I
T
H
F
I
N
G
E
R
!
Figure 7.1: Maximizing Height