Lesson - Bounce (Projectile motion and collisions)

Gravity is a force which causes any mass to attract any other mass. On the surface of the Earth, the largest object nearby is the earth itself, and all objects on the Earth's surface are held onto the Earth by the Earth's gravitational pull.

While the gravitational force on any two masses near the surface of the Earth does not need to be the same (and is in fact larger for larger masses), the acceleration caused by that force is the same for all objects near the Earth's surface, even if we are considering a bowling ball and a BB.

The force on any object near the Earth's surface due to the Earth is given by F=mg, where g is the acceleration due to gravity. At sea level, this acceleration is approximately 9.8 m/s².

To consider the action of gravity on an object, consider a ball which rolls off of a table and bounces on the floor.

The ball in the bounce experiment will undergo primarily three phases: resting on a table, free fall, and bouncing. The fundamental forces during each of these phases are:

• Resting on Table:
  • Gravity
  • Normal Force
• Free Fall:
  • Gravity
• Bouncing:
  • Gravity
  • Elastic force (ball acts as a spring)

Resting on Table:

While the object is resting on the table, it does not move up or down, that is, it's velocity in the y direction is zero the entire time. If the velocity is not changing, then it's acceleration is also zero.

By Newton's Second Law, we see that if the acceleration of the object is zero, the net force on the object is also zero.

Free Fall:

When the object is in free fall, only gravity acts on the object (neglecting air resistance).

Bouncing:

When the object bounces, it compresses as it hits the table. Assuming the table does not compress, the force given by the compression is equal to the spring constant of the object times the compression, and is in the direction opposite the compression, F=-kx. In the case of the ball falling on the table, the table presses back against the ball in an upwards direction. This is similar to the normal force, except that the compression of the ball results in a force greater than the weight of the ball, and it is propelled upwards.

Using data from the web or from an experiment of your own construction, measure the position of a ball at differnt times as it rolls off a table and onto the floor. Calculate the first and seconf derivative of the position and answer the following questions.

Questions:

1. Describe the forces that act at each stage of motion:
   1. During which parts of the experiment does gravity act on the ball?
   2. During which parts of the experiment is there a force on the ball due to the table?
   3. During which parts of the experiment is there a force on the ball due to the floor?
   4. During which parts of the experiment is there a force on the floor due to the ball?
   5. During which parts of the experiment is there a force on the table due to the ball?
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.
3. 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?
4. What is the net force on the ball while it is falling down? due to gravity? due to the table? due to the floor?
5. What is the average net force on the ball for the duration of the time that it bounces? How much of this is due to the table? the floor? gravity?
6. What is the net force on the ball after the bounce, but while it is still moving up? How much of this is due to the table? due to the floor? due to gravity?

Advanced Questions:

1. Can we trust the "edge" data points between stages of motion?
2. What resolution (i.e. number of data points per second) is required to make a reasonable interpretation?
3. Estimate the uncertainty in measurement. How does measurement error in position and time affect computational error in the velocity and acceleration?