# Projectile Motion

Updated 08/2014 gmn

## Materials

1 each ball launcher w/ball and loading plunger

1 each large ring stand and base

1 each meter stick / measuring tape

masking tape

plain white paper, and carbon paper

C-clamp

## Goals

1) Derive an equation for the range of a projectile fired at an arbitrary launch angle and height.

2) 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.

3) Predict, then measure and compare the range of a projectile when the projectile is fired at an arbitrary angle with respect to the horizontal.

4) Predict the initial firing angle of the launcher for a prescribed range value. Then compare the range achieved with the desired range value.

## Introduction

When objects undergo motion in two (or even three) dimensions rather than in just one, the over-all 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 the Earth experiences a constant acceleration. An object moving horizontally has zero acceleration. Any other choice of perpendicular directions would have different constant values of acceleration in both of the directions. When we write the descriptions of the motion in mathematical terms, the horizontal/vertical choice of directions results in the simplest forms of the equations.

Under what conditions can the effects of the air be ignored? One condition is that the speed is not too high, since the effect of the air increases with the 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 (for example, a ping-pong ball and a steel ball bearing of the same size). In this lab care has been taken to make these choices so that ignoring air resistance will have a very small effect on the trajectory of the projectile.

## Exercise 1: Mathematical Preliminaries

To accomplish the first two of our stated goals we need first to find a general mathematical relationship showing how the horizontal range of the projectile depends on the height, initial velocity, and the angle of launch. See Fig. 1..

You need to solve symbolically the appropriate kinematics equations for motion with constant acceleration in the horizontal and vertical directions simultaneously to do this. [Hint: Eliminate the time from the general horizontal and vertical position equations at the instant when the projectile strikes the floor! You can do this by solving one equation for time and substituting that expression in the other equation.]

Rather than writing the equations with the angle, θ, explicitly shown, it is suggested that you use the symbols v_{ox} and v_{oy} where v_{ox} = v_{o} cosθ and v_{oy} = v_{o} sinθ to simplify the algebraic manipulations. You need to solve for the range, R, in terms of v_{ox}, v_{oy}, h, and g. In your final expression, you can substitute v_{o} and θ back in, since these will be measured directly later on.

** The details of this derivation must be shown in your lab report.** Check your final formula with your lab instructor before proceeding to Exercise 2.

Fig. 1.

Fig. 2. **Picture of Ball Launcher**

**Instructions and Precautions for Using the Ball Launcher:**

__Warning: Never look down the barrel of launchers. Wear safety glasses for the duration of the lab.__

#### Step 1) Make sure that the launcher is attached securely to the table so it does not move when the launcher is fired. A C-clamp will secure the ring stand base to the table. Make sure the launcher is at the proper angle by using the built-in plumb bob located on the side of the launcher. Note that the angle measured by this plumb bob is actually the same as the angle between the “barrel” of the launcher and horizontal.

#### Step 2) Since the projectiles will be hitting the floor, use the additional plumb bob to locate and mark the position on the floor (masking tape helps) 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 easily be measured later. You’ll have to measure the height to get the vertical distance. Be sure to show carefully on a diagram between which two points the height measurement is made. If you are unsure, please discuss it with your lab instructor.

#### Step 3) To launch the projectile, load the ball into the projectile launcher. Use the plunger 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 gently straight up (perpendicular to the launch tube actually) on the string attached to its trigger on the top.

Step 4) To record the position where the projectile strikes the floor, tape a provided white paper target to the thin hard board sheet (about 0.3 m x 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. The ball will leave a dark smudge on the white paper as it bounces. If necessary you can tape the hard board sheet to the floor to keep it from moving, but please avoid the indiscriminate use of masking tape on the floors. **If you do put tape on the floor, please remove it when you are finished.**

## Exercise 2: Determining the Initial Speed (“Muzzle Velocity”) of the Projectile

- Simplify the general equation that you found in Exercise 1 for the case when θ = 0 (horizontal launch). Then solve for v
_{o}in terms of R, h, and g.

- Set the launcher to fire horizontally, that is, a launch angle of zero degrees.

- Pick one of the range settings, and fire the projectile (using the four steps in the previous section) a couple of times noting where the projectile lands. Place the paper target centered as best you can where the ball will land. Now use the carbon paper to record permanently the landing position of four launches at the same initial conditions.

- From each data point on the range, R, determine the magnitude of the initial velocity of the ball. Calculate the average of these velocity values. speed of the ball as it was launched.

- Calculate the standard deviation of the measured speed values for the launcher setting you chose. This will represent your uncertainty in measuring the launch speed, which will be used in Exercise 5. (Refer to the Uncertainty/Graphical Analysis Supplement near the back of the lab manual and Exercise 5 for more details – or ask your lab instructor.)

## Exercise 3: Predicting/Measuring the Landing Position at Angles Other Than Zero

- Choose a launch angle between 30 and 40 degrees. Using the values of the initial speed of the ball as determined in Exercise 2 and your general equation for R from Exercise 1, calculate the horizontal distance (range) from the launch point to where the ball should land at the range setting you chose in Exercise 2 for the initial launch angle that you have chosen.

- In turn place a paper target on the floor at each of the calculated positions and fire the projectile. If the projectile misses the target completely, check your calculations and/or discuss it with your lab instructor. If the projectile does hit the target, then repeat several times to get a good average experimental range value with its corresponding standard deviation to compare with your calculated range. Compare the predicted range values with the experimental ranges values within the uncertainties given by the respective standard deviations. Percent differences between the average experimental range values and the predicted values may also be useful.

## Exercise 4: Predicting Launch Angle for a Desired Range

- Ask your lab instructor for an assigned value of horizontal distance (range) for your group.

- Your task is to calculate a suitable angle (or angles in some cases) at the range setting you used in Exercises 2 & 3. 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:

**tan θ = v_{0}^{2}/gR + [(v_{0}^{2}/gR)^{2} – 1 + 2v_{0}^{2}h/gR^{2}]^{1/2}**

where *v*_{0} is the initial launch speed, R is the range, h is the initial launch height, and g is the acceleration of gravity. Show your calculations in your final report. You can earn up to 10 extra credit points by showing the details of the calculation to get this algebraic expression for tan θ. [Hint: Again you should start with the horizontal and vertical position equations.]

- Now set the target and do the experiment with your lab instructor present to observe. Were you able to hit the target? If you have trouble here, check your calculations. Is your calculator in radian or degree mode? Get assistance from your lab instructor, if necessary. As before compare your experimental range with appropriate uncertainties to the range value assigned by your lab instructor.

## Exercise 5: Uncertainty Analysis Check List [Refer to the Uncertainty/Graphical Analysis Supplement at the back of the lab manual – or ask your lab instructor]

__Exercise 2:__

In Exercise 2, you measured the horizontal distance, or range, values over a few different trials. Since these distances did not all come out exactly the same for a given launch condition, some uncertainty is associated with the measurement. A rather obvious “best” distance value is simply the average, or mean, of the distances, but how do we get a numerical value for the uncertainty associated with it? The sample standard deviation, denoted by σ_{x}, is a way to do this for repeated values that we think should really be the same because we did not change any of the conditions. The ball is initially launched at some height above the floor, and this measurement also has some uncertainty determined by the measuring instrument and our measurement technique.

Both of these uncertainties contribute to the uncertainty in the initial speed of the ball. Thus we must “propagate” the uncertainty of both these measurements to get the uncertainty in the value of the initial speed. The “maximum-minimum” method is appropriate for PHYS 101, but the “partial derivative” method is appropriate for PHYS 201. Show your calculations clearly in your report.

__Exercise 3:__

In Exercise 3, you calculated theoretical values for the range of the projectile assuming that the initial speed, initial height, and the launch angle were known quantities. However, each of these quantities has some uncertainty associated with it. Thus, your prediction for R cannot be expected to be 100% accurate. As with the v_{0} value you calculated in Exercise 2, You will need to propagate the uncertainties in q, v_{0}, and h, in order to find the uncertainty in your predicted range value: R ± dR.

The measured ranges are also “fuzzy” numbers, having uncertainty associated with them as you have previously calculated using the standard deviations. If the “fuzziness” of the calculated range values overlaps with the “fuzziness” (s_{R}) of the experimental values, then you can claim they are the same within the uncertainty of the experiment. If not, then you must conclude that they did not agree, and you may wish to resort to a percent difference as a means of comparison.

## Conclusion

Summarize all your results, preferably in a table where the quantities with their uncertainties and comparisons between predicted values and experimental values are clearly displayed. Are you convinced that the theoretical predictions made by separating the horizontal and vertical motions agree with experiment, at least within the calculated uncertainties of the experiment? Your answers need to be based on your experimental results and the calculated uncertainties of the quantities you are comparing. Please don’t make vague statements that are not directly supported by your calculations and measurements.

Before you leave the lab
Straighten up your lab station. Make sure that the projectile ball is back in the plastic tray. Report any problems or suggest improvements to your lab instructor. |