A2L Item 062

• Description: Identify correct statements about an elastic collision between two bodies involving a spring between them.
• Goal: Interlink several dynamical concepts and associate with a physical process.
• Source: UMPERG-ctqpe98
• Keywords: Mechanics, Interactions, Energy, Springs

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The question for students:

Mass M₁ is traveling along a smooth horizontal surface and collides with a mass M₂ (stationary) which has a spring attached as shown below.

The spring between the blocks is most compressed when

1. all the energy in the system is potential energy stored in the spring.
2. the net momentum of the system is zero.
3. the velocity of the center of mass has its smallest value.
4. mass M₁ is no longer delivering an impulse to Mass M₂.
5. the only kinetic energy in the system is that of the center of mass.
6. none of the above
7. cannot be determined

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Commentary for teachers:

Answer

5: When the spring is maximally compressed, both masses have the same velocity which is the velocity of the center of mass.

Background

The total energy of an isolated system can be decomposed into three categories; kinetic energy associated with center of mass motion, kinetic energy of bodies in the center of mass coordinate frame and potential energy associated with the interaction of bodies comprising the system.

Questions to Reveal Student Reasoning

Is M₂ ever traveling faster than M₁?

Do the masses ever have the same velocity?

How would you find P_cm, the momentum of the center of mass?

How is the kinetic energy of the center of mass related to its momentum?

Suggestions

A sketch of the velocities of the two masses over time would look something like the graph at the right. [Note that the velocity of M₁ can be negative after the collision if it is less massive than M₂.] Such a graph helps make the relationships between V_cm, the relative velocity of the masses and the spring compression clear.

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