PROJECTILE MOTION
Exp. #5
90 minutes
(Alward/Harlow web File: “projecti.doc” 1-20-04)
Name: _____________________ Partners: _______________________Section No. _______
Equipment:
Projectile Launcher, Ramrod, Plastic Ball,
Photogate Timer with Photogate
Time of Flight Accessory + Extension Cord, Meter Stick, Masking Tape (10"),
1/4 Sheet Carbon Paper, 1/4 Sheet White Paper, Masking Tape, Table Clamp, AC Adapter
I. PURPOSE
The purpose of this exercise is to study the motion of a projectile. The range and flight time of a
projectile launched from a table top and striking the floor will be calculated and measured.
To predict where a ball will land on the floor when it is shot off a table at some angle above the
horizontal, it is necessary to first determine the firing angle and the initial speed. The firing angle is
easily measured using the protractor-plumb bob assembly attached to the projectile launcher.
Measurement of the initial speed is discussed below:
II. MEASURING INITIAL SPEED
1. Clamp the projectile launcher to the corner of the table and adjust the level of the launcher so the ball
will be shot off horizontally. Insert a yellow ball into the launch tube and cock it to the long range
position.
Fire one shot to locate where the ball hits the floor. At this
position, tape a piece of white paper to the floor. Place a
piece of carbon paper (carbon-side down) on top of this
paper and tape it down. When the ball hits the floor, it will
leave a mark on the white paper.
2. Fire about ten shots.
3. Estimate by eye the approximate center of the cluster of
ten marks. Measure the distance between the cluster center
to the point on the floor directly below the launch position
mark on the side of the launcher. Record this horizontal
distance, d in the Table I.
1
4. Measure the distance h in meters between the bottom of
the figure of the ball drawn on the side of the launcher, to the
floor directly below. Record this distance in Table I.
The vertical position of a projectile as a function of time is given by Equation 1 below:
y = y0 + (v0 sin )t - ½ gt2
(1)
where y0 is the initial altitude at time t = 0,  is the angle above horizontal at which the projectile
was fired, and v0 is the initial speed. In the present case, y0 = h, and  = 0 since the projectile is
fired horizontally. At the end of the flight, at time t, the projectile strikes the ground (y = 0).
Thus,
0 = h - ½ gt2
(2)
5. Using Equation 2 above, use the measured value of h and the value g = 9.8 m/s2 to calculate
the flight time. Show your work below and record the time in Table I.
Calculate flight time from Equation (2):
t=
s
Report this value below in Table I.
6. The initial velocity of the projectile, v0, is the same as its initial horizontal velocity, since the
initial vertical component of velocity is zero. Now, since the horizontal velocity does not change
during the flight, the average horizontal velocity is the same as v0 :
v0 = d/t
(3)
2
Table 1: Measuring Initial Velocity
Horizontal Distance
d=
m
Elevation
h=
m
Flight Time
(calculated above)
t=
s
Initial Velocity
v0 = d/t =
m/s
II. PREDICTING THE FLIGHT TIME OF A BALL FIRED AT AN ANGLE
1. Adjust the angle of the projectile launcher to an angle between
30 and 60 degrees and record this angle in Table II.
2. Measure the vertical distance between the top of the time-offlight plate and the bottom of the ball’s launch position (marked
on the front side of the launch tube). This distance should be
about 1-2 cm less than the elevation recorded earlier, because the
height of the strike plate on the ground subtracts from the vertical
distance found in Part I. Record this distance h in Table II.
The projectile starts out at an initial elevation y0 = h. The ycoordinate of the ball’s position at any instant of time t is given by
Equation 1. Note: since the ball is fired upward, it will take longer
to strike the ground than it does if it is fired straight ahead, as in
Part I. Thus, the "t" used in the equations below is not the same "t"
used in Part I. If we let this new "t" be the total flight time to the
ground (where y = 0), Equation 1 becomes
0 = h + (v0 sin)t - ½ gt2
(4)
The equation above is quadratic in the variable time variable, and it may be re-written as
½ gt2 - (v0 sin)t - h = 0
(5)
3
Recall that the general quadratic equation is
at2 + bt + c = 0
(6)
whose solutions are
t = [-b ± (b2 - 4ac)1/2]/2a
(7)
Use Equation 7 to solve Equation 5 for the projectile’s predicted flight time, t. Show your work
below and report the result in Table II.
Solve:
a=
½ gt2 - (v0 sin)t - h = 0
½g =
Note: v0 is the value obtained earlier, shown in Table I.
b = - v0 sin =
Show work here:
4
c=-h=
TABLE II
Firing Angle
=
deg
Elevation
(not the same h in Table I)
h=
m
Calculated Flight Time
(from Equation 5)
t=
s
Measured Flight Time
s
1.
2.
3.
4.
5.
Average Measured Flight Time:
t=
s
Percentage Difference:
(between average measured and
the predicted flight time)
%
4. Put the plastic ball into the launcher and cock it to the long range position. Fire one shot to
locate where the ball hits the floor. Tape the the time-of-flight plate at this position. Tape a
sheet of white paper to the top of the plate, and on top of the paper place a piece of carbon paper.
5. Attach support bar to launcher. Detach the
timer’s photogate and attach it to the support bar,
close to the launcher tube's opening.
Connect the time-of-flight cord to the timer and set
the timer to “pulse”, 1 ms. Connect the voltage
adapter. The timer will start when the ball breaks
the infra-red light beam, and stop when the ball
strikes the plate.
6. Fire five shots and record the flight times in Table II
5
7. Calculate the percentage difference between the predicted flight time recorded earlier, and the
average measured flight time in Table II. The percentage difference is the absolute value of the
difference between the predicted and measured times, divided by the predicted value, times 100.
-
III. PREDICTING THE RANGE OF A BALL FIRED AT AN ANGLE
The "range" of a projectile fired over level ground is just the horizontal distance traveled. This
horizontal distance, labeled x, is given by Equation 8 below:
x = (v0 cos) t
(8)
1. Using Equation 8 above, calculate the expected range of the ball. Note that the predicted
range is just the constant horizontal velocity of the ball, times the predicted flight time. Show
your calculation below and record the range in Table III.
Calculation of the range of the projectile using Equation 8 (use predicted time, t):
x=
m
2. Locate and mark the approximate center of the cluster of the four impact marks on top of the
strike plate. Measure the distance between this center to a point on the ground directly below the
ball’s launch position. Record this measured distance in Table III.
3. Calculate the percentage difference between the predicted and measured values of the ball’s
range. This percentage is found by taking the absolute value of the difference between the
values, dividing by the predicted value, and multiplying by 100.
6
Table 3: The Range
Predicted Range:
(from Equation 8)
x=
m
Measured Range:
x=
m
Percentage Difference:
%
When you have finished, do not throw away the carbon paper.
7