- #1

uSee2

- 57

- 7

- Homework Statement
- Two small carts, Cart 1 of 5 kg and Cart 2 of 10 kg are set in contact with each other at position x = 0 on a larger platform. The platform is centered on position x = 0 and is 24 meters long. The carts and platform can roll on bearings of negligible friction and are all intially at rest.

At time t = 0, a small spring between the two carts expands, sending Cart 1 to the left with an intiial speed of 4 m/s. Both carts collide and bounce of the bumpers at the ends of the platform, which can be assumed to be perfectly elastic. When the carts collide with each other again, they stick together.

Suppose the platform's wheels are free to roll, but the platform has much more mass than the two carts. If the experiment is repeated again exactly as before, will the center of the platform be to the left, to the right, or at x = 0 when the carts collide again and stick? Explain.

- Relevant Equations
- ##p = mv##

My Explanation:

This system is a closed system, so the center of mass velocity stays constant. It was initially at rest so the position of the center of mass is constant. After their collision, the 2 carts are to the right of x = 0. Center of mass originally was at x = 0, so the platform had to move to the left to balance it out to keep the center of mass in the same position. As such, the platform is to the left from x = 0.

I believe that my explanation above is correct, I could be wrong though. If my explanation is correct, then couldn't there be any case in which the platform moves to the right or left?

If ##m_p## is the mass of the platform, right is positive, and since momentum initially equals zero, conservation of momentum can be written like:

##p_i = 0##

##p_f = p_i = 0##

##p_f = 0= m_2v_2 - m_1v_1 \pm m_pv_p##

Since ##v_1## and masses are known it can be written as:

##0 = 10v_2 - (5)(4) \pm m_pv_p##

##0 = 10v_2 - 20 \pm m_pv_p##

Here is where I'm sorta confused. It states that ##m_p## is really big, so shouldn't ##m_pv_p## approach infinity? Even if ##v_p## gets smaller, ##m_p## would be large so it would be still large. I know that this definitely doesn't happen in real life, since if it did that would mean ##10v_2 - 20## approaches infinity in the opposite direction, so cart 2 would just fly away.

My questions:

##v_2## has to be positive, right? It cannot be negative in the 2nd experiment as that means both carts would move to the left.

What makes ##m_pv_p## approach zero rather than infinity if ##m_p## is really large?

Also, are we allowed to assume cart 1 moves at 4 m/s in the 2nd experiment as well? If so, why?

This system is a closed system, so the center of mass velocity stays constant. It was initially at rest so the position of the center of mass is constant. After their collision, the 2 carts are to the right of x = 0. Center of mass originally was at x = 0, so the platform had to move to the left to balance it out to keep the center of mass in the same position. As such, the platform is to the left from x = 0.

I believe that my explanation above is correct, I could be wrong though. If my explanation is correct, then couldn't there be any case in which the platform moves to the right or left?

If ##m_p## is the mass of the platform, right is positive, and since momentum initially equals zero, conservation of momentum can be written like:

##p_i = 0##

##p_f = p_i = 0##

##p_f = 0= m_2v_2 - m_1v_1 \pm m_pv_p##

Since ##v_1## and masses are known it can be written as:

##0 = 10v_2 - (5)(4) \pm m_pv_p##

##0 = 10v_2 - 20 \pm m_pv_p##

Here is where I'm sorta confused. It states that ##m_p## is really big, so shouldn't ##m_pv_p## approach infinity? Even if ##v_p## gets smaller, ##m_p## would be large so it would be still large. I know that this definitely doesn't happen in real life, since if it did that would mean ##10v_2 - 20## approaches infinity in the opposite direction, so cart 2 would just fly away.

My questions:

##v_2## has to be positive, right? It cannot be negative in the 2nd experiment as that means both carts would move to the left.

What makes ##m_pv_p## approach zero rather than infinity if ##m_p## is really large?

Also, are we allowed to assume cart 1 moves at 4 m/s in the 2nd experiment as well? If so, why?