1. Elastic and Inelastic Collisions

2. What do you think?
Collisions are sometimes described as elastic or inelastic. To the right is a list of colliding objects. Rank them from most elastic to most inelastic.
What factors did you consider when ranking these collisions?
A baseball and a bat
A baseball and a glove
Two football players
Two billiard balls
Two balls of modeling clay
Two hard rubber toy balls
An automobile collision
When asking students to express their ideas, you might try one of the following methods.
(1) You could ask them to write their answers in their notebook and then discuss them.
(2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups.
The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally.
Answers will vary, with the toy balls generally near the top of the list and the football players near the bottom of the list. The point is to get students to think about the meaning of these terms in an everyday context before hearing the definitions. Some students may realize that billiard ball collisions are nearly perfectly elastic. Listen to their ideas and try to see if there is any consensus among the students without interjecting the “correct” answers. In order to determine an exact order for the list, we would need to measure the loss in KE for each of these collisions.

3. Perfectly Inelastic Collisions
Two objects collide and stick together.
Two football players
A meteorite striking the earth
Momentum is conserved.
Masses combine.
Final velocity is the same for both objects
Ask students to suggest additional examples of perfectly inelastic collisions.
m1v1,i + m2v2,i = (m1 + m2) vf

4. Classroom Practice Problems
An 2.0 x 105 kg train car moving east at 21 m/s collides with a 4.0 x 105 kg fully-loaded train car initially at rest. The two cars stick together. Find the velocity of the two cars after the collision.
Answer: 7.0 m/s to the east
Momentum is conserved, and that is the basis for the first problem. (2.0 x 105 kg)(21 m/s) = (6.0 x 105 kg) (v)
Students should find that KE is not conserved. In fact, it is reduced significantly.

5. Classroom Practice Problems
An 2.0 x 105 kg train car moving east at 21 m/s collides with a 4.0 x 105 kg fully-loaded train car initially at rest. The two cars stick together.
Calculate the kinetic energy of the two cars before and after the collision. Was kinetic energy conserved?
Answer: KEbefore= 4.4 x 107 J, KEafter= 1.5 x 107 J
KE is not conserved. It is less after the collision.
Momentum is conserved, and that is the basis for the first problem. (2.0 x 105 kg)(21 m/s) = (6.0 x 105 kg) (v)
Students should find that KE is not conserved. In fact, it is reduced significantly.

6. Inelastic Collisions Kinetic energy is less after the collision.
It is converted into other forms of energy.
Internal energy - the temperature is increased.
Sound energy - the air is forced to vibrate.
Some kinetic energy may remain after the collision, or it may all be lost.
If the objects are still moving after the collision, then there is still some KE. If both objects have stopped, such as might occur in some head-on collisions, then all of the KE is converted into other forms of energy.

7. Elastic Collisions m1v1,i + m2v2,i = m1v1,f + m2v2,f
Objects collide and return to their original shape.
Kinetic energy remains the same after the collision.
Perfectly elastic collisions satisfy both conservation laws shown below.
Momentum and Kinetic Energy
m1v1,i + m2v2,i = m1v1,f + m2v2,f
Billiard balls colliding are nearly perfectly elastic. Ideal gases undergo perfectly elastic collisions between molecules and between the walls of the container.
m1v1,i2 + m2v2,i2 = m1v1,f2 + m2v2,f2

8. Elastic Collisions
Two billiard balls collide head on, as shown. Which of the following possible final velocities satisfies the law of conservation of momentum?
vf,A = 2.0 m/s, vf,B = 2.0 m/s
vf,A = 0 m/s, vf,B = 4.0 m/s
vf,A = 1.5 m/s, vf,B = 2.5 m/s
Answer: all three
m = 0.35 kg m = 0.35 kg
v = 4.0 m/s v = 0 m/s
This slide and the next slide are designed to allow students to see that conservation of momentum can occur in several ways. However, conservation of momentum and kinetic energy can only occur in one way. This will take some time, but should develop the idea that conservation of both KE and momentum limits the possible results in an elastic collision. See the next slide for further explanation.
Momentum before the collision = 1.4 kg•m/s.
Any of the three results shown yields a total momentum after the collision of 1.4 kg•m/s.

9. Elastic Collisions
Two billiard balls collide head on, as shown. Which of the following possible final velocities satisfies the law of conservation of kinetic energy?
vf,A = 2.0 m/s, vf,B = 2.0 m/s
vf,A = 0 m/s, vf,B = 4.0 m/s
vf,A = 1.5 m/s, vf,B = 2.5 m/s
m = 0.35 kg m = 0.35 kg
v = 4.0 m/s v = 0 m/s
This slide and the previous slide are designed to allow students to see that conservation of momentum can occur in several ways. However, conservation of momentum and kinetic energy can only occur in one way. This will take some time, but should develop the idea that conservation of both KE and momentum limits the possible results in an elastic collision.
KE before the collision is 1/2(0.35 kg)(4.0 m/s)2 = 2.8 J.
The first choice for velocities produce a KE after the collision of 1/2(0.35 kg)(2.0 m/s)2 + 1/2(0.35 kg)(2.0 m/s)2 = 1.4 J (a decrease in KE).
Similarly, the last data points also show a decrease in KE.
Only the velocities of 0 m/s for A and 4.0 m/s for B produce no change in KE. Therefore, this is the only possible result if the collision is elastic.
An infinite number of possible velocities would satisfy the conservation of momentum. The only result possible in a perfectly elastic collision (for equal masses) is that one ball stops and the other continues with the original speed.
Many students have seen the sets of 5 stainless steel balls hanging from threads and allowed to swing back and forth as they collide. If one ball is dropped, only one goes up from the other side. This device behaves like it does because the collisions are nearly perfectly elastic. The steel balls are deformed very slightly, and spring back with little loss of energy.
Answer: only vf,A = 0 m/s, vf,B = 4.0 m/s

10. Types of Collisions
Recap the three types of collisions. Ask students to describe the ”what happens” and “conserved quantity” columns before you reveal the text.