Lesson Plan - Bounce (Projectile motion and collisions)

The Bounce exercise is aimed at reinforcing concepts in Newtonian mechanics, kinematics, and projectile motion. The goal is to have the student analyze the motion of an object through a complex path, first rolling, then falling, then bouncing, and falling again. By doing this, the student must answer questions related to what forces are acting at a given time.

AP Physics curriculum goals:

• Motion in two dimensions
• Projectile motion

National Standards:

• Science Content Standards: 9-12, CONTENT STANDARD A:
• Use technology and mathematics to improve investigations and communications.
• Formulate and revise scientific explanations and models using logic and evidence.
• Recognize and analyze alternative explanations and models.
• Communicate and defend a scientific argument.
• Science Content Standards: 9-12, CONTENT STANDARD B:
• Motions and forces

Numerical Methods:

• Numerical Integration
• integrate acceleration to get velocity
• integrate velocity to get position

Misconceptions:

• Acceleration is the same as velocity.
• The acceleration of a falling object depends upon its mass.
• Freely falling bodies can only move downward.
• Gravity only acts on things when they are falling.

Suggested Answers to Questions:

1. Describe the forces that act at each stage of motion:
   0. During which parts of the experiment does gravity act on the ball? Gravity acts on the ball at all times.
   1. During which parts of the experiment is there a force on the ball due to the table? When the ball is rolling across the table
   2. During which parts of the experiment is there a force on the ball due to the floor? When the ball is bouncing, and after it has stopped.
   3. During which parts of the experiment is there a force on the floor due to the ball? Whenever there is a force on the ball due to the floor.
   4. During which parts of the experiment is there a force on the table due to the ball? Whenever there is a force on the ball due to the table.
2. You notice that during the time that the ball is on the table, the acceleration and velocity are both zero. Your colleague points out that this is obvious, as when the velocity is zero the acceleration must also be zero. How do you respond? Back up your response with data from your experiment. The acceleration is the rate of change of the velocity. If the velocity is zero and remains zero, it is constant and therefore acceleration is also zero, such as the point at which the ball is rollingon the table. If the velocity is zero, but changing, such as the moment the ball bounces, or at the top of the ball's arc, the velocity is zero for a moment, but the acceleration is clearly non-zero.
3. What is the net force on the ball during the time that it is rolling on the table? Zero. When the ball is rolling, gravity and the normal force of the table balance each other out. What is the force due to the table? Equal in magnitude to gravity, but pointed up. due to the floor? Zero. The floor is not in contact with the ball. due to gravity? F_g=mg, pointed towards the floor.
4. What is the net force on the ball while it is falling down? According to our data a ~ 10 m/s², and ma = 1 N downwards. due to gravity? Since the force due to the table and floor are contact forces, and no contact is made, only gravity acts during free fall, and F_g = ma = 1 kg m/s² downwards. due to the table? Zero. due to the floor? Zero.
5. What impulse is delivered to the ball on the first bounce? Is it possible to determine the force that acts on the ball during the bounce? For our experiment, the impulse was ~ (0.1 kg) * ( +3.0 m/s). The instantaneous force is a very sudden event, and data needs to be taken throughout the bounce to track the force through the event. If we could determine at least the duration of the bounce, we could at least determine an average force, but our data does not permit us to determine the duration of the bounce. We can, however, place a lower limit on the average force by using as the duration of the bounce our time step.
6. What is the net force on the ball after the bounce, but while it is still moving up? After the bounce. only gravity acts on the ball, and F_g= mg = 0.1 kg * 10 m/s² = 1 N downwards. How much of this is due to the table? None. due to the floor? None. due to gravity? All.

Advanced Questions:

0. Can we trust the "edge" data points between stages of motion? No, the problem with the "edge" points is that we have no record of when the object actually made contact or left contact. We only have evidence that it occurred somewhere before or after. In some cases, it is even difficult to determine whether contact is currently being made, and there is a range of two intervals instead of one in which we cannot determine when contact occurred or ceased.
1. What resolution (i.e. number of data points per second) is required to make a reasonable interpretation? This depends upon the event in question. If you merely want to estimate the acceleration of an object, little resolution is required. Our model with data taken every 0.08 seconds was able to get to within 10 %. Given the uncertainty in accurately placing the exact position of the ball, this is likely as good as we can hope for. However, we clearly cannot resolve the time dependance of the force due to bouncing with time resolution less than the duration of the bounce.
2. Estimate the uncertainty in measurement. Given our images, the ball position and exact center was uncertain due to shadowing or having a stretched image during the exposure, and has uncertainty on the order of a few percent in position. How does measurement error in position and time affect computational error in the velocity and acceleration? Using the rand() function in excel, 5% or greater error introduced into either our position or time data started to show qualitatively different results.

Suggested Alternate Exercise:

Construct "hypothetical" data for the motion of an object, such as a falling body. Using a computer, a set of dice, or some method of generating random numbers, look at how adding "error" into the solution affects the numerical results. What are the limitations of these methods depending on the solution to be anyalyzed?