Hello, I'm trying to figure out which of the following statements are TRUE.

I suspect that 2, 6, 8 are true...and that there is one more that is true, but not sure which one?
Thanks!

It is impossible to have a collision between two objects in which the center-of-mass of the system is stationary the entire time.


If the magnitude of the momentum of one object is smaller after the collision than it is before the collision, the magnitude of the momentum of the other object must be larger after the collision than it is before the collision.


The vector sum of the momenta of the two objects before the collision equals the vector sum of the momenta of the two objects after the collision.


Momentum and kinetic energy are always conserved.


Momentum and kinetic energy are conserved only when the collision is elastic (when the elasticity equals 1).


The magnitude of the change in momentum for one object in the collision is equal to the magnitude of the change in momentum for the other object in the collision.


The motion of the center-of-mass is completely unaffected by any change in the elasticity of a collision.


Momentum is always conserved, but kinetic energy is conserved only when the collision is elastic.


If the center-of-mass of the system is moving in one direction before the collision it must move in the opposite direction after the collision.

To determine which of the statements are true, let's go through each statement and explain how to evaluate its truthfulness:

1. It is impossible to have a collision between two objects in which the center-of-mass of the system is stationary the entire time.

To determine the truth of this statement, we need to understand what a collision is and how it affects the center-of-mass. A collision occurs when two objects interact and exert forces on each other. The center-of-mass is the average position of the system, considering the masses and positions of the objects.

In a collision, the center-of-mass can move as a result of the forces involved. However, it is not impossible for the center-of-mass to remain stationary during a collision. For example, if two objects of equal mass collide head-on with equal but opposite momenta, the center-of-mass will remain stationary. Therefore, statement 1 is false.

2. If the magnitude of the momentum of one object is smaller after the collision than it is before the collision, the magnitude of the momentum of the other object must be larger after the collision than it is before the collision.

The principle of momentum conservation states that the total momentum of an isolated system remains constant before and after a collision, assuming no external forces act on the system.

If one object's momentum decreases, then the other object's momentum must increase to compensate for the change in total momentum. Therefore, statement 2 is true.

3. The vector sum of the momenta of the two objects before the collision equals the vector sum of the momenta of the two objects after the collision.

The principle of momentum conservation also implies that the vector sum of the momenta before the collision is equal to the vector sum of the momenta after the collision. This principle holds regardless of the type of collision (elastic or inelastic).

Therefore, statement 3 is true.

4. Momentum and kinetic energy are always conserved.

This statement is not entirely correct. Momentum is always conserved, as stated in statements 2 and 3, but kinetic energy is not always conserved, especially in inelastic collisions where energy is lost.

Therefore, statement 4 is false.

5. Momentum and kinetic energy are conserved only when the collision is elastic (when the elasticity equals 1).

This statement is an extension of statement 4. As mentioned, momentum is always conserved, but kinetic energy is only conserved during elastic collisions, where there is no loss of energy.

Therefore, statement 5 is true.

6. The magnitude of the change in momentum for one object in the collision is equal to the magnitude of the change in momentum for the other object in the collision.

According to the principle of momentum conservation, the total change in momentum for one object is equal in magnitude but opposite in direction to the total change in momentum for the other object.

Therefore, statement 6 is true.

7. The motion of the center-of-mass is completely unaffected by any change in the elasticity of a collision.

The motion of the center-of-mass is independent of the elasticity of a collision. Changes in elasticity may affect the motion of individual objects involved in the collision, but the overall motion of the center-of-mass remains unaffected.

Therefore, statement 7 is true.

8. Momentum is always conserved, but kinetic energy is conserved only when the collision is elastic.

This statement is a summary of statements 4 and 5. It restates that momentum is always conserved, while kinetic energy is only conserved in elastic collisions.

Therefore, statement 8 is true.

9. If the center-of-mass of the system is moving in one direction before the collision, it must move in the opposite direction after the collision.

The motion of the center-of-mass of a system is affected by the forces exerted during a collision. If there is an external force acting on the system or if the collision is not perfectly symmetric, the motion of the center-of-mass may not reverse direction.

Therefore, statement 9 is false.

In summary, the true statements among the given options are 2, 3, 5, 6, 7, and 8.