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 momentumof 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).
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 Netwon'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 = (Vf + Vi) * t
Vf = Vi + at
d = Vi * t + ½ * a * t2
Vf2 = Vi2 + 2 * a * d
Where d is the displacement of an object, Vi is the initial velocity of the object, Vf 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 Vi = 0 m/s a = 9.81 m/s2 Vf = ? t = ?
d = Vi * t + ½ * a * t2
¬1 m = 0 m/s * t + ½ * 9.81 m/s2 * t2
t = 0.45 s
Vf2 = Vi2 + 2 * a * d
Vf2 = 0 m/s + 2 * 9.81 m/s2 * 1 m
Vf = 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.
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.
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.
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.
Could you play tennis with a baseball or soccer with a basketball? (Listen to student responses.) What are all the different sports that are played with balls? (Possible answers: Volleyball, soccer, football, softball, baseball, ping pong, wiffle ball, bowling, dodge ball, golf, jacks, tennis, croquet, raquetball, squash, tetherball, etc.) What are some differences and similarities among the balls used for different sports?
How do the materials and design of a ball affect its characteristics? A soccer ball is designed to be bouncy, flexible and full of air, making it great to be kicked down a soccer field without injuring players. A bowling ball is dense, heavy and hard so that it can be rolled down a bowling alley to hopefully get a strike rather than a gutter ball. Each ball is designed with specific materials, making it appropriate for a particular sport.
When engineers are given a design task, whether it is designing a new volleyball that can bounce twice as high or a new airplane or skyscraper, they must study and analyze the properties of the materials they would like to use. What might be some material properties that they consider ? (Possible answers: Weight, strength, hardness and flexibility.)
Do you think it is important to understand materials and their properties, especially in the design of a ball used in a game? Well, imagine being the goalie in a soccer game that uses a bowling ball instead of a soccer ball. OUCH!!!
This activity coincides well with math graphing practice.
Description of different graph types (line, scatter, bar, pie). Nice example pictures. https://www.keynotesupport.com/excel-basics/excel-chart-types.shtml
This is a link to an online game that teaches mean, median, and mode. http://www.kidsmathgamesonline.com/numbers/meanmedianmode.html
Allows children to create graphs and experiments with probability. https://nces.ed.gov/nceskids/createagraph/
- Gather materials and make copies of the worksheets.
- Explain the two tests that will be done to determine the bouncing properties of different balls.
- Divide the class into groups of three students each. One student serves as the recorder, one drops the ball, and one is the timekeeper.
- Assign each group a ball. After running both tests on that ball, have the groups switch balls (rotate) and test a new ball until all balls have been tested by each group.
- Conduct tests and record data.
Test 1: Ball Bounce Height Comparison
The first time you drop the ball do not take a measurement, just watch where the ball goes so the next time the observer knows where to look. This help to greatly increase the accuracy of the experiment. Drop a ball from 1 foot off of the floor, slightly in front of a yardstick. Measure the height the ball reaches after the first bounce and record. Repeat this test from 2 ft, 3 ft, and 1/2 ft. Do this test for each ball and record data. To increase accuracy, you may repeat each test three times and divide by 3 to find an average.
Test 2: Ball Bounce Time Comparison
Drop a ball from a height of 3 ft, timing from when the ball is released until the ball stops bouncing. Record the time. Repeat this test for each ball. Talk with the students about coming up with a system for releasing the ball and starting the stop watch. Possible suggestions are to have the same student drop the ball and start the watch, or have the two students count down from five.
- Graph group results. (If this activity is not able to be accompanied by a math lesson on graphing, introduce the topic before the activity starts or perhaps after the class has recorded its data and worked through it as a group.)
- Compare results as a class.