Bouncing Balls: Collisions, Momentum & Math (for High School)

Sports engineering is becoming a popular specialty field of study. While some engineers dedicate their research to understanding collisions between balls and bats, others study the effects of a golf ball colliding with the head of a golf club. And, mechanical engineers consider momentum and collisions when designing vehicles. Learning how the human body and equipment interacts with the ball during impact or how the human body interacts with the inside of a car during a crash, helps engineers design better sports equipment and safer vehicles.

After this activity, students should be able to:

• Understand that momentum depends on both mass and velocity.
• Recognize that different surfaces and materials promote different types of collisions.
• Collect data to solve equations.
• Learn that understanding material properties is an important aspect of engineering design

Momentum can be thought of as mass in motion and is given by the expression:

Momentum = mass x velocity

The amount of momentum an object has depends both on its mass and how fast it is going. For example, a heavier object going the same speed as a lighter object would have greater momentum. Sometimes when moving objects collide into each other, momentum can be transferred from one object to another. There are two types of collisions that relate to momentum: elastic and inelastic.

An elastic collision follows the Law of Conservation of Momentum, which states "the total amount of momentum before a collision is equal to the total amount of momentum after a collision." In addition, the total kinetic energy of the system (all the objects that collide) is conserved during an elastic collision. An elastic collision example might involve a super-bouncy ball; if you were to drop it, it would bounce all the way back up to the original height from which it was dropped. Another elastic collision example may be observed in a game of pool. Watch a moving cue ball hit a resting pool ball. At impact, the cue ball stops, but transfers all of its momentum to the other ball, resulting in the hit ball rolling with the initial speed of the cue ball.

In an inelastic collision, the total momentum of the system is conserved, but the total kinetic energy of the system is not conserved. Instead, the kinetic energy is transferred to another kind of energy such as heat or internal energy. A dropped ball of clay demonstrates an extremely inelastic collision. It does not bounce at all and loses its momentum. Instead, all the energy goes into deforming the ball into a flat blob.

In the real world, there are no purely elastic or inelastic collisions. Rubber balls, pool balls (hitting each other), and ping-pong balls may be assumed extremely elastic, but there is still some bit of inelasticity in their collisions. If there were not, rubber balls would bounce forever. The degree to which something is elastic or inelastic is dependent on the material of the object (see Figure 1).

Figure 1. The illustration of the percentage of energy returned to a ball after one bounce (or, the "range of bounciness" of each ball).CopyrightCopyright The Exploratorium © 2007, http://www.exploratorium.edu/baseball/howfar7.html

Another way to understand collisions is through Newton's 3rd Law, which tells us that "for every action, there is an equal and opposite reaction". When a cue ball collides with another pool ball, the cue ball exerts a force on the stationary pool ball in the direction that the cue ball is traveling, while the stationary pool ball exerts an equal and opposite force on the cue ball. This is the reason that after the cue ball collides with a stationary pool ball, it sometimes moves in a new direction, sometimes leading to a "scratch". Understanding Newton's 3rd Law, momentum and elastic and inelastic collisions provides a new understanding of our physical world that is full of motion and collisions.

In order to complete this activity, you will also need to have an understanding of the motion of an object. Following are the Kinematics equations:

d = (V_f + V_i) * t

V_f = V_i + at

d = V_i * t + ½ * a * t²

V_f² = V_i² + 2 * a * d

Where d is the displacement of an object, V_i is the initial velocity of the object, V_f is the final velocity, a is the acceleration of the object, and t is the interval of time the object traveled. For example, if a ball is rolled off of a table 1 meter above the ground, we can find the velocity with which it hits the floor and the time it takes to do so:

d = 1 m V_i = 0 m/s a = 9.81 m/s² V_f = ? t = ?

d = V_i * t + ½ * a * t²

¬1 m = 0 m/s * t + ½ * 9.81 m/s² * t²

t = 0.45 s

V_f² = V_i² + 2 * a * d

V_f² = 0 m/s + 2 * 9.81 m/s² * 1 m

V_f = 4.43 m/s

If we have three known values, then we must choose equations that use the three values that actually we do have to find the ones that we do not. You also have to read between the lines sometimes to get three known values. For example, in the problem stated previously, the value of acceleration is not given but the object is in free fall, meaning its acceleration is that of gravity.

Before the Activity

• Gather materials.
• Make enough copies of the Bouncing Balls Worksheet so that each student has one.

With the Students

1. Determine the mass in kilograms of each ball and record it on the data sheet.
2. Drop each ball from a distance of 1 meter onto the surface and record how high it bounces in meters (example: 0.46 meters).
3. Note whether the ball and surface showed more of an elastic or inelastic collision.

• If the ball bounces up more than .5 meters, then it is more elastic.
• If it bounces up less than .5 meters, then it is more inelastic.

4. Repeat steps 1, 2 and 3 for the two other surfaces.
5. Calculate the velocity for each ball right before it bounces (question 2) and right after it bounces (question 3).
6. Calculate the momentum for each ball right before it bounces (question 4) and right after (question 5).
7. Calculate the percentage of momentum lost for each case (question 6).
8. Answer the Further Learning questions on the worksheet based on your answers. (Note: Have students complete question 11 as a group.)
9. Once the class is finished with the Bouncing Balls Worksheet, discuss which balls had the best elastic collisions on each surface. Also, if time permits go over some of the Further Learning questions as a class.

Pre-Activity Assessment

Brainstorming: In small groups, have the students engage in open discussion. Remind students that no idea or suggestion is "silly." All ideas should be respectfully heard. Ask the students:

• What are sports examples of transfer and conservation of momentum? (Possible answers: Hitting a baseball with a bat, hitting the cue ball with a pool stick, the cue ball bouncing off another ball, striking a golf ball with a club or driver, or hitting a tennis ball with a racquet.)

Activity Embedded Assessment

Voting: Ask the students to vote to rank the sports (named above) from those having the greatest momentum to those having the least momentum. While the students will have to use their own judgment, remind them that momentum depends equally on mass and velocity.

Post-Activity Assessment

Problem Solving: Present the class with the following cases:

• Case 1: A big-time slugger hits a baseball 60 meters/sec (134 mph).
• Case 2: Johnny knocks down four pins at the Bowl-a-Rena by rolling a 15-pound bowling ball 1.34 meters/sec (3 mph).

Ask students which ball would bounce higher if each were thrown onto a trampoline with the given velocities. What about on concrete? (Answer: The bowling ball would bounce higher on the trampoline, while the baseball would bounce higher off of concrete.)

Discuss as a class why this is the case. Notice that the trampoline responds with a higher bounce to objects of greater mass, while the concrete causes objects with greater elasticity to bounce higher.

Be sure the students do not use the balls as projectiles.

This activity is best done in groups, because while one person drops the ball, another person must watch the ball and meter stick to note how high the ball bounces. Additional team members could hold the meter stick steady and/or record the data. It is difficult to get an accurate measurement for how high the ball bounces since it is in constant motion. Therefore, have students drop each ball on each surface several times, or until they have a consistent measurement.

Some balls are greatly affected by wind resistance, such as wiffle balls. Therefore, try to pick balls that will not have much influence from wind resistance since this experiment is done under the assumption there exists no wind resistance.

If students have never seen the kinematics equations, this can be a good introduction. Help the students figure out the exact equations they will need to use and walk them through the parts of the worksheets that involve the kinematics equations.

The Physics Classroom and Mathsoft Engineering & Education, Inc., 2004, accessed May 30, 2007. http://www.physicsclassroom.com/Class/momentum/momtoc.html

Momentum and energy loss of balls colliding against different surfaces, accessed May 30, 2007. http://www.iit.edu/~smile/ph8709.html

The Exploratorium, Science of Baseball, accessed May 30, 2007. http://www.exploratorium.edu/baseball/index.html

The Exploratorium, Science of Baseball, accessed May 30, 2007. http://www.exploratorium.edu/baseball/howfar7.html

The Exploratorium, Science of Baseball, accessed May 30, 2007. http://www.exploratorium.edu/baseball/howfar5.html