I was going over some conceptual problems in physics and got confused by how ans

Question

I was going over some conceptual problems in physics and got confused by how answers to 1 and 2 seemingly contrdict the answer to 3. If someone could explain these concepts it would be much appreciated.

1. Q: Consider two carts, of masses m and 2m, at rest on an air track. If you push first one cart for 3 s and then the other for the same length of time, exerting equal force on each, the momentum of the light cart is

1. A: Equal to the momentum of the heavy cart.

2. Q: Consider two carts, of masses m and 2m, at rest on an air track. If you push first one cart for 3 s and then the other for the same length of time, exerting equal force on each, the kinetic energy of the light cart is

2. A: Larger than the kinetic energy of the heavy cart.

3. Q: Two objects are sitting at rest on a “frictionless” air hockey table. Object A has twice the mass of object B. Both objects are pushed with the same force for the same distance. Which statement is true?

3. A: The objects have different momentum but the same kinetic energies.

For A. 3, I would have thought that it was oppisite. That they would have the same momentum but different kinetic energies. Why is this not the case?

Explanation / Answer

1 . Impulse = Force x time = Change in Momentum. Since Force and time are same, therefore Momentum is same for both objects.

2. Similarly , same Impulse, therefore same momentum, but, since K.E. = momentum ^2 / 2m, therefore light cart mass has more K.E.

For the 3rd Answer:

Since Work Done is same for both objects ( Work Done = force x Displacement), therefore, applying Work Energy Theorem:

Work Done = Change in K.E.

Therefore Kinetic Energy for both objects are same.

Now, Since Momentum = Sqrt ( 2m x K.E.)

Therefore, both objects will have different momentum, since their masses are different, though K.E. is same

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