PHY 105 Investigation on Projectile Motion
Introduction
The purpose of this investigation is to verify the equations of projectile motion. This purpose will be
accomplished in two parts. First, using the equations of projectile motion, determine the initial velocity of a
horizontally launched small plastic sphere fired from a spring-launched mechanism. Second, assuming this
initial velocity is independent of firing angle, calculate the range, using the equations of projectile motion,
required when the launcher is set to a specified launch angle θ2. If the calculated range and actual range are
within the expected error, the equations of projectile motion will be verified.
y
y
v0
v0
θ2
y2i
y1i
x
x1f
x
x2f
Equations of Projectile Motion
Horizontal Motion – Constant Velocity
Vertical Motion – Constant Acceleration
xf  xi  v0 cos   t
yf  yi  v0 sin    t 
1
g  t 2
2
Combining Horizontal and Vertical Motions for
Combining Horizontal and Vertical Motions for
  0 and y f  0
  0 and y f  0

 xf
 1  xf

yf  yi  v0 sin    
  g 

 v 0 cos   2  v 0 cos  
yf  yi  tan    xf 
2
gx
0  yi   f2
2v 0
 v0 
g
 xf
2 yi
2
gxf
2
2v 0 cos 2  
2
 xf
 1  xf

yf  yi  v0 sin    
 g


 v 0 cos   2  v 0 cos  
yf  yi  tan    xf 
2
gxf
2
2v 0 cos 2  
2
gx
0  yi  tan    xf  2 f 2
2v 0 cos  

 2
g
 xf  tan  xf  yi  0
  2
2
 2v 0 cos   
2
Procedure
1) Set the launcher to a firing angle of 1  0 (i.e. horizontal). Measure and record the initial position of the
small projectile y1i where it leaves the launcher relative to the point on the floor directly under the launcher.
2) Fire the projectile horizontally several times, recording where it impacts the floor. Measure and record the
final average final x-position x1f . Estimate and record the uncertainty in the average final x-position x1f .
3) Calculate the initial velocity of the projectile v0 using the equations of projectile motion. Calculate the
uncertainty in the initial velocity v0 . Record both these parameters.
4) Set the launcher firing angle to that values specified by your laboratory instructor and record this launch
angle θ2.
5) Calculate the predicted range x2,Calc of the projectile launched at θ2 and record this value.
6) Calculate and record the expected uncertainty in the predicted x-position x2,Calc assuming that the only
significance source of error is the uncertainty in the initial velocity v0 .
7) Test the equations of projectile motion by firing the projectile several times at the launch angle θ2. Record
the mean final x-position x2, Actual and the uncertainty in x2, Actual . If the actual final x-position x2, Actual and the
calculated final x-position x2,Calc are within each other’s uncertainty, they you may consider that the
equations of projectile motion have been verified.
8) If the actual final x-position x2, Actual and the calculated final x-position x2,Calc are not within each other’s
uncertainty, they the equations of projectile motion have not been verified and you should attempt to explain
why these time tested equations failed to produce the actual measured range.
Structure of Laboratory Report


Title Page
Introduction


Purpose
How the purpose will be
accomplished
Data and calculations
State your conclusion
regarding the purpose.
This handout
Notes on calculating the uncertainties
v 0
and x2f .
For the case where   0 and y f  0
Using calculus principles, calculate
dv0
:
dx
v0 
2g
 xf
yi
dv0
2g

and
dx
yi

dv0 v0

dx xf
or
2 g v0

yi
xf
dv0 dx

v0
xf
If we replace the “d”s with “Δ”s in the derivatives we arrive at
x will result in a 5% error in v0 (i.e.
 v 0 x
x

 5% error in
. So a, for example
v0
xf
xf
v0
 5% ).
v0
For the case of   0 and y f  0
Calculate the expected uncertainty in its final x-position x2f due to the uncertainty in the velocity v0 . Since
x2 f is a non-linear function of velocity, rather than use a difficult derivative calculation, just evaluate the
uncertainty in x2 f numerically by solving the quadratic equation

 2
g
 2
 xf  tan  xf  yi  0
2
 2v 0 cos   
explicitly for the two values of x2 f from the two extreme values of velocity v0  v0 and v0  v0 . This will
give the extreme values for the final x-position. The difference in the extreme values of x2 f is x2f . You can
then express the predicted final position as x2f  x2f