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Projectile Motion: Determining Launch Speed and Angle for Given Range, Lab Reports of Physics

Washington State University (WSU or Wazzu)Physics

A lab experiment aimed at determining the launch speed and angle of a projectile to achieve a prescribed range. Instructions on how to mathematically derive the relationship between range, launch height, initial velocity, and angle, as well as how to measure the range experimentally using a ball launcher. The document also includes exercises to calculate the initial speed, predict landing positions at angles other than zero, and perform uncertainty analysis.

Typology: Lab Reports

Pre 2010

Uploaded on 08/30/2009

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(version 3/24/06)
Projectile Motion
Goals
1) Determining 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.
2) Being able to predict and measure and compare the range of a projectile when the projectile is
fired at an arbitrary angle with respect to the horizontal.
3) Being able to 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. We know this by experiment. Likewise 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. When conditions are such that air resistance cannot be
ignored, the motion is much, much more complicated.
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 on the next page. 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 vox and voy where vox = vo
cosθ and voy = vo sinθ to simplify the algebraic manipulations. You need to solve for the range,
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Download Projectile Motion: Determining Launch Speed and Angle for Given Range and more Lab Reports Physics in PDF only on Docsity!

(version 3/24/06)

Projectile Motion

Goals

  1. Determining 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.

  2. Being able to predict and measure and compare the range of a projectile when the projectile is fired at an arbitrary angle with respect to the horizontal.

  3. Being able to 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. We know this by experiment. Likewise 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. When conditions are such that air resistance cannot be ignored, the motion is much, much more complicated.

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 on the next page. 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 vox and voy where vox = vo cosθ and voy = vo sinθ to simplify the algebraic manipulations. You need to solve for the range,

R, in terms of vox, voy, h, and g. The details of this derivation must be shown in your lab report.

Fig. 1

Picture of Ball Launcher

Fig. 2

Instructions and Precautions for Using the Ball Launcher:

y v 0

θ

h

x R

  1. Calculate the standard deviation of the measured range values for each of the three launcher range settings. Using the uncertainties in the range and launch height values, calculate the uncertainties in the initial launch speed for each range setting. (Please refer to the Uncertainty/Graphical Analysis Supplement near the back of the lab manual and Exercise 5 for more details.)

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

  1. 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 short, medium, and long range launcher settings for the initial launch angle that you have chosen.
  2. 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 TA. 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: The Real Test

  1. Ask your TA for an assigned value of horizontal distance (range) for your group.
  2. Your task is to calculate a suitable angle (or angles in some cases) 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:

tan θ = v 02 /gR + [( v 02 /gR)^2 - 1 + 2 v 02 h/gR^2 ]1/

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 5 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.]

  1. 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 here, check your calculations. Is your calculator in radian or degree mode? Get assistance from your TA, if necessary. As before compare your experimental range with appropriate uncertainties to the range value assigned by your TA.

Exercise 5: Uncertainty Analysis Check List [Refer to the Uncertainty/Graphical Analysis Supplement at the back of the lab manual]

In Exercise 2 you measured at least three horizontal distance, or range, values for each launcher setting (short, medium, or long range). 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 σx, 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.

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. The measured ranges are “fuzzy” numbers, in other words having uncertainty associated with them as you have previously calculated using the standard deviations. If the calculated range values lie within the “fuzziness” 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.

In Exercise 4 again compare your given range value with the “fuzzy” experimental range value to determine whether they agree or not. Use a percent difference as a backup.

In 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 TA.