Exp. 7: vary the angle to maximize the height, Purpose, Theory – PASCO ME-6831 Ballistic Pendulum_Projectile Launcher User Manual
Page 39
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M o d e l N o . M E - 6 8 3 0
E x p . 7 : V a r y t h e A n g l e t o M a x i m i z e t h e H e i g h t
0 1 2 - 0 5 3 7 5 C
35
Exp. 7: Vary the Angle to Maximize the Height
Equipment Needed
Purpose
The purpose of this experiment is to find the launch angle that will maximize the height on a vertical wall for a
projectile launched at a fixed horizontal distance from the wall.
Theory
When the ball is shot at an angle at a fixed distance, x, from a
target such as a vertical wall, the ball hits the wall at a height y
given by:
where y
0
is the initial height of the ball, v
0
is the initial speed
of the ball as it leaves the muzzle,
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
Solving this equation for the time of flight, t, gives
Substituting for t in the equation for y gives
To find the angle,
, that gives the maximum height, y, find the first derivative of the equation for y and set it equal
to zero. Solve for the angle,
.
Solving for the angle,
, gives:
Since the second derivative is negative for
max
, the angle is a maximum. To find the initial speed of the ball, use
the fixed distance, x, and the maximum height, y
max
. Solve the y-equation for v
0
and plug in the values for y
max
,
max
, and x.
Item
Item
Projectile Launcher and plastic ball
Board to protect wall
Meter stick or measuring tape
Sticky tape
White paper, large sheet
Carbon paper (several sheets)
Plumb bob and string
θ
CA
UTIO
N!
DO
NO
T LO
OK
DO
WN BAR
RE
L!
CA
UTIO
N!
DO
NO
T LO
OK
DO
WN BAR
RE
L!
CAUTION!
DO NOT LOOK
DOWN THE
LO
NG
RA
NG
E
ME
DIU
M
RA
NG
E
SH
OR
T
RA
NG
E
Po
sit
ion
of
Ba
ll
La
un
ch
SHOR
T RANGE
PR
OJ
EC
TIL
E L
AU
NC
HE
R
ME-6800
Ye
llo
w B
an
d i
n W
ind
ow
Ind
ica
tes
Ra
ng
e.
9
8
7
6
5
4
3
2
1
0
WEAR
SAFETY
GLASSES
WHEN IN USE.
Us
e 25 m
m
ba
lls
ON
L
Y!
Figure 7.1: Maximizing Height
Range
Initial
height
Angle
Initial
speed
y
y
0
v
0
x
y
y
0
v
0
sin
t 1
2
---gt
2
–
+
=
x
v
0
cos
t
=
t
x
v
0
cos
-----------------
=
y
y
0
x
gx
2
2v
o
2
2
cos
-----------------------
–
tan
+
=
dy
d
------
x
2
gx
2
2
sec
tan
v
0
2
----------------------------------
–
sec
0
=
=
max
tan
v
0
gx
------
2
=