Lab 2. Projectile Motion

Goals

• To determine the launch speed of a projectile and its uncertainty by measuring how far it

travels horizontally before landing on the ﬂoor (called the range) when launched horizontally

from a known height.

• To predict and measure the range of a projectile when the projectile is ﬁred at an arbitrary

angle with respect to the horizontal.

• To predict the initial ﬁring angle of the launcher for a prescribed range value.

• To determine quantitatively whether the measured ranges in (2) and (3) are consistent with

the desired range values.

Introduction

When objects undergo motion in two (or even three) dimensions rather than in just one, the overall

motion can be analyzed by looking at the motion in any two (or three) mutually perpendicular

directions and then putting the motions “back together,” so to speak. In the case of projectiles, the

horizontal and vertical directions are usually chosen. Why is this choice made? Ignoring the effects

of air resistance, an object moving vertically near the surface of Earth experiences a constant ac-

celeration. We know this by experiment. Likewise an object moving horizontally experiences zero

acceleration. Any other choice of perpendicular directions would have nonzero, constant values

of acceleration in both directions. When we write the descriptions of the motion in mathematical

terms, the horizontal/vertical choice of directions results in the simplest description.

Under what conditions can the effects of air resistance be ignored? One condition is that the

object’s speed is not too high, since the effect of the air resistance increases with speed. If two

objects are the same size and shape, the lighter one of the two will experience the larger effect on

its motion due to the air. (Imagine a ping-pong ball and a steel ball bearing of the same size.) In

designing this lab, care has been taken to ensure that air resistance has a negligible effect on the

trajectory of the projectile. When conditions are such that air resistance cannot be ignored, the

motion is more complicated.

16 CHAPTER 2. PROJECTILE MOTION

Mathematical preliminaries—Equation for range

To accomplish the ﬁrst two of our stated goals, we need a general mathematical relationship be-

tween the horizontal range of the projectile and the initial height, initial velocity, and launch angle.

See Figure 2.1. You will need to solve the appropriate kinematics equations for motion with con-

stant acceleration in the horizontal and vertical directions simultaneously. Rather than writing the

equations in terms of the angle, θ, it is suggested that you use the symbols v

and v

, where

= v

cosθ and v

= v

sinθ , to simplify the algebra. You need to solve for the range, R, in

terms of v

, v

, h, and g. The details of this derivation must be included in your lab notes.



Figure 2.1. Coordinate system for calculating the range, R.

Instructions and precautions for using the ball launcher

Warning: Never look down the barrel of a launcher. Wear eye protection until all the groups

have ﬁnished launching projectiles.

1. Make sure that the launcher is attached securely to the table so it does not move when the

launcher is ﬁred. Make sure the launcher is at the proper angle by using the built-in plumb

bob on the side of the launcher. Note that the angle measured by this plumb bob is the angle

between the “barrel” of the launcher and the horizontal.

2. Since the projectiles will be hitting the ﬂoor, use a second plumb bob to locate and mark

the position on the ﬂoor (blue tape works) directly below the launch point of the projectile.

This indicates the initial horizontal position of the ball at ﬂoor level so the range (horizontal

distance traveled by the ball) can be measured later. You will have to measure the height to

get the vertical distance. Clearly indicate in a diagram how you measured the height (from

where to where). If you are not sure how the height should be measured, please discuss it

with your TA.

3. To launch the projectile, load the ball into the projectile launcher. Use the rod to push the

ball into the launch tube to one of the three preset launch positions (short, medium, or long

range). You will hear a click as you reach each position. Notify others nearby and across

the room before ﬁring the ball. Stand out of the way and ﬁre the launcher by pulling on the

string attached to its trigger on the top. To minimize the force applied by the string to the

launch tube, pull the string at right angles to the launch tube.

4. To record the position where the projectile strikes the ﬂoor, tape a white paper target to the

thin hard-board sheet (about 0.3 m × 0.5 m in size) at your lab station. Place the sheet and

target at the approximate place where the ball lands. When you are ready to record some

landing points, lay a piece of carbon paper (carbon side down) on top of the target. Do not

put tape on the carbon paper. The ball will leave a dark smudge on the white paper where it

lands. If necessary you can tape the hard-board sheet to the ﬂoor to keep it from moving, but

avoid the indiscriminate use of tape on the ﬂoors.

Determining the initial speed of the projectile

1. Simplify your general equation for the range for the case when θ = 0 (horizontal launch).

Then solve for v

in terms of R, h, and g.

2. Set the launcher to ﬁre horizontally, that is, to launch at an angle of zero degrees. Care with

this angle setting can signiﬁcantly improve your results later in the lab.

3. Starting with the medium range launch setting, ﬁre the projectile (using the four steps in the

previous section) a couple times noting where the projectile lands. Center the paper target

as best you can where the ball will land. Now use the carbon paper to record the landing

position of four or ﬁve launches using the same initial conditions.

4. From your data determine the average range, R, of the ball. Use this average distance to

calculate the average initial speed of the ball as it was launched.

5. Repeat the same procedure for the short and long range settings on the launcher.

Range for nonzero launch angles

1. Choose a launch angle between 30° and 40°. Using the values of the initial speed of the ball

measured above and your general equation for the range, calculate the horizontal distance

(range) from the launch point to where the ball should land for the short and medium range

settings using the initial launch angle that you have chosen. (Do not use the long range

setting.)

2. For the short and medium range settings, place a paper target on the ﬂoor at the calculated

position and ﬁre the projectile. If the projectile misses the target completely, check your

calculations and/or discuss it with your TA. If the projectile does hit the target, then repeat

several times to get a good average experimental range value and its corresponding standard

deviation to compare with your calculated range.

3. Compare your predicted range values with the experimental range values using your ex-

perimental standard deviations. Assume that your predicted range, R

predicted

has zero un-

18 CHAPTER 2. PROJECTILE MOTION

certainty. Then your measurement is consistent with your prediction if t

= |R

measured

−

predicted

|/σ(R

measured

) < 3. If you ﬁnd that t

> 3, check your calculations and consider

carefully what systematic errors may be present in your experiment.

Launch angle to achieve a given range

1. Ask your TA to assign a value of horizontal distance (range) for your group.

2. Calculate a suitable angle at one of the range settings for launching the projectile to the target

set at the assigned distance. The relationship giving the initial launch angle in terms of the

other parameters is:

tanθ =





−1 +

1/2

(2.1)

3. Now set the target and do the experiment with your TA present to observe. Were you able to

hit the target? If you have trouble, check your calculations. Is your calculator in radian or

degree mode? Get assistance from your TA, if necessary. Again, compare your experimental

range value to the range value assigned by your TA. If not, check your calculations and your

procedure.

Conclusion

Summarize all your results, preferably in a table showing the measured and calculated quantities

with their uncertainties. Clearly display your comparisons between predicted values and experi-

mental values. Are you convinced that the theoretical predictions made by separating the horizon-

tal and vertical motions agree with experiment, at least within the calculated uncertainties of the

experiment? Your answers must be based on your experimental results and the calculated uncer-

tainties of the quantities you are comparing. Do not make vague statements that are not directly

supported by your calculations and measurements.

Before you leave the lab please:

Return the projectile and the carbon paper to the TA Table.

Remove all tape from the ﬂoor.

Wrap the plumb bob string around the cardboard spool.

Store the plumb bob and string in its plastic bag.

Return the goggles to their plastic bags.

Place the plumb bob, tape measure, goggles and rulers in your equipment basket.

Straighten up your lab station.

Report any problems or suggest improvements to your TA.