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Lab 4. Projectile Motion, Lecture notes of Kinematics

International University Institute of Luxembourg (IUIL)Kinematics

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.

Typology: Lecture notes

2021/2022

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Lab 4. Projectile Motion
Goals
ā€¢ 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.
ā€¢ To predict and measure the range of a projectile when the projectile is fired at an arbitrary
angle with respect to the horizontal.
ā€¢ To predict the initial firing angle of the launcher for a prescribed range value.
ā€¢ 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 ā€œback together,ā€ 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
objectā€™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.
25
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Lab 4. 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 ā€œback together,ā€ 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 objectā€™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.

26 CHAPTER 4. PROJECTILE MOTION

Mathematical preliminariesā€”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 4.1. 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, Īø , it is suggested that you use the symbols v 0 x and v 0 y, where v 0 x = v 0 cos Īø and v 0 y = v 0 sin Īø , to simplify the algebra. You need to solve for the range, R, in terms of v 0 x, v 0 y, h, and g. The details of this derivation must be included in your lab notes.

e

v 0

h

R x

y

Figure 4.1. 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 ā€œbarrelā€ 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

28 CHAPTER 4. PROJECTILE MOTION

certainty. Then your measurement is consistent with your prediction if tā€²^ = |Rmeasured āˆ’ Rpredicted |/Ļƒ (Rmeasured ) < 3. If you find that tā€²^ > 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:

tan Īø =

v^20 gR

[(

v^20 gR

2 v^20 h gR^2

] 1 / 2

  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, 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.