PASCO ME-6825A MINI LAUNCHER - Manual (Page 33)

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
cos
θ)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
2v
0
2 cos 2 q
To find the angle that gives the maximum height, y, set dy/d
θ equal to zero and solve for the angle.
y = y
0
+ v
0
sin q t - 12 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
90 80 70 60 50 40 30 20 10 0 10 20 30 40
MI NI
LA UN
CH ER
ME-6825 CA UTION!
WE AR SA FE TY
GL AS SE S
WH EN IN US E.
DO N'T PU SH
PIS TO N WI TH
FIN GE R!
Figure 7.1: Maximizing Height