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1 Projectile motion

The purpose of the experiment is to either perform a computer assisted velocity measurement, or to perform an indirect velocity measurement, and to experimentally test the equations for projectile motion.

The apparatus consists of a muzzle-loading spring gun, plunger, plastic ball, a vernier caliper and two-meter stick or tape measure, and optionally a PC running Pasco Science Workshop, and a pair of photo-gates.

1.1 The range and trajectory

The trajectory traced out by a projectile launched from the surface of the earth is a parabola (in the absence of air resistance) gotten by eliminating

t

between the pair of kinetic equations

The range

R

is the horizontal distance traveled by the projectile before it hits the ground, which in the ﬁgure is at

y = - h

. The range then is the solution to

1.2 Experimental procedure

The ﬁrst task is to determine the “muzzle velocity”

v_{0}

of the spring gun. The projectile used in this experiment is a ball shot from a spring gun. The direct determination of initial speed

v_{0}

can be made by using a computer to measure the passage time of the ball through a photo-gate as it leaves the gun. Use the vernier caliper to measure the diameter

D

of the ball.

1.2.1 Computerized velocity measurement

Pasco Science Workshop couples an analog/digital data acquisition box to the serial or parallel port of the computer. The box can be connected in turn to a variety of sensors and other devices, such as photo-gates, smart pulleys, meters, and counters.

In this experiment we will measure the velocity of the projectile by timing its passage between two photo-gates.

The photo-gate measures the amount of time that a beam of light is blocked by the passage of a body through the beam. There is a small red LED on the gate that indicates when the beam is broken.

To correctly conﬁgure the Pasco interface for this experiment, take the following steps;
1. Log onto the lab computer at your station, and double click in the “Shortcut to Sciwkshp” icon to start Pasco Science Workshop.

2. Click and drag the plug to digital channel 1 on the interface. This will open the Digital Sensor Window.

4. The timing of the ball through the gate must be compensated for the width or diameter of the ball. Measure the ball diameter. It should be

0.025 m

. Set the Object Length to this value. Now place the two photo-gates a distance

0.25 m

apart, and set the Spacing Between Gates to this value.

5. Set the interface to display the photo-gate data by clicking on and dragging the Digits display icon (there are Digits, Meter, Scope, FFT, Table, and Graph icons) on the Science Workshop window to digital channel 1.

6. This action will open a calculations menu, from which you should select Velocity Between 1&2, v1t02 [m/s] and click on Display.

This action will open up a digital display such as that illustrated below. Double click in the window to set or reset the digits. You are now ready to record data.

7. To record data, click on the record button (REC) in the upper left corner of the Science Workshop main window. Press stop when you are ﬁnished collecting data.

Shoot the ball through the gate ﬁve times, and record the velocity for each run. Average these and compute a standard deviation. This is the muzzle velocity or initial speed

v_{0}

for the experiment.

1.2.2 Range Measurements

Once the speed

v_{0}

has been established, the ﬁnal part of the experiment is to predict where the ball should hit the ground if launched at angles

θ_{0} = 0

and

θ_{0} = 30

degree angles above the horizontal. The gun has an attached protractor for precise angle-of-elevation measurement.
Very carefully measure

h

, the elevation from the ﬂoor to the muzzle of the gun before each set of launchings. Fire the gun eight times for each of the two elevation angles, and measure

R

for each shot. The geometry is fully described in the ﬁrst ﬁgure of the experiment. Taping a piece of paper to the ﬂoor with a carbon paper over it will make a better record of the impact point.
Determine whether or not the results agree with the prediction made from your formula, and compute the percent error.

1.3 Pre-lab questions

1. If you threw a ball straight up in the air, using “muzzle velocity”

v_{0} = 20.0 \frac{m}{s}

, what would have been the maximum height that it would have reached?

2. Consider the apparatus used in the experiment (shoot a spring gun from a table top). Neglecting air resistance, calculate the balls speed when it strikes the ﬂoor for

θ_{0} = 0

, using

h = 1.0 m

and muzzle velocity

v_{0} = 5.0 \frac{m}{s}

3. Show that if air resistance is neglected, the maximum range that a gun can shoot (over level ground) occurs when it is aimed at a

4 5^{o}

angle, and that this range is

R_{m a x} = \frac{v_{0}^{2}}{g}

4. There are two launch angles

θ

at which a gun will shoot

50 %

of its maximum possible range (

R_{m a x} = \frac{v_{0}^{2}}{g}

) over level ground. Find these two launch angles.

1.4 Lab report

Projectile Motion


Experimenter 1		Experimenter 2

Experimenter 3		Experimenter 4

Determination of $v_{0}$

If you measured the ball diameter

D

and the transit time through the beam, ﬁll out the ﬁrst row. If you made a direct velocity measurement, ﬁll out the second.

Transit times

Transit speeds






$t_{1} (s e c)$	$t_{2} (s e c)$	$t_{3} (s e c)$	$t_{4} (s e c)$	$t_{5} (s e c)$	$\bar{t}$	$σ_{t}$










$v_{1} (m ∕ s e c)$	$v_{2} (m ∕ s e c)$	$v_{3} (m ∕ s e c)$	$v_{4} (m ∕ s e c)$	$v_{5} (m ∕ s e c)$	$\bar{v}$	$σ_{v}$

Range measurements and calculations

Range Measurements






Trial	1	2	3	4	5	6	7	8	${\bar{R}}_{e x p}$


$θ_{0} = 0^{o}$



$θ_{0} = 3 0^{o}$

Experimental and theoretical comparison






Launch angle	${\bar{R}}_{e x p}$	$σ_{R}$	$R_{t h e o r y}$	$% e r r o r$



$θ_{0} = 0^{o}$



$θ_{0} = 3 0^{o}$

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