How do I find the impulse and final speed in this problem?

Question

I am revising for my physics exam and I got stuck on this problem:

A trolley with mass 100kg is moving along a track at 7m/s (ignore friction). We drop a 20kg box from a bridge above the track. Right before it lands, it is travelling straight down at 5m/s. The trolley has an opening, that opens some time after the box has landed into the trolley. What is the impulse on the tracks after the box has fallen into the trolley? What is the velocity of the trolley after the box falls through it?

I've tried using the formula for impulse J=pf−pi, but the initial momentum is 0 because the tracks aren't moving. Therefore because of conservation of momentum, the final momentum is also 0. But then the impulse is 0 and that doesn't make any sense...

For the second part, I've calculated the velocity after the box falls into the trolley to be 5.83m/s, but then due to conservation of momentum, the momentum before and after the box falls through it should be the same. The momentum before it falls is pi=700kgm/s. This is then equal to 100kg∗vf, so the final velocity is the same as initial velocity, which also doesn't make sense to me.

Where did I go wrong?

Answer 1

First part - This is tricky because momentum is a vector quantity. In this case there are both vertical and horizontal momenta. The vertical momentum (being considered in the first part) is NOT conserved because there are outside, unbalanced forces acting on the system vertically. So, they want you to calculate the impulse on the box (mass times change in velocity of the box) to determine (by Newton's 3rd Law) the impulse on the tracks (don't forget direction). Again, since there are other forces on the track, this momentum is not conserved.

Second part - Since we can ignore friction, there are no outside horizontal forces acting on the system, so the horizontal momentum WILL be conserved. Consider ONLY the horizontal momenta (non-zero for the cart and zero for the bag initially), and use your equation to solve for the final horizontal momentum of the system to find the final velocity of the trolley/box.