Finding the Mistake in Falling Objects in an Elevator

[llandau]

1. An elevator is moving upward with constant velocity V. We consider a frame of reference fixed to the elevator. Since it is not accelerating, our system is equivalent to, say, a frame of reference fixed to the ground (there are no inertial forces). Now, we are inside the elevator and let an object fall from a height h. The only force acting is gravity. If there is an elastic collision with the floor, the object will reach the same height h, as we expected.

2. If v=-gt is the velocity of the object relative to the ground, the velocity for an observer inside the elevator must be v'=-gt-V. So, x'=-(1/2)gt^2-Vt. At t=0, x'=h. We find that, when x'=0, v'=-sqrt(V^2+2hg). After the collision, the object will start moving upward with velocity sqrt(V^2+2hg) but it seems to me that it will reach a height less than h: h+V^2/g-(V/g)sqrt(V^2+2hg).

There must be a mistake somewhere. Can you help me finding it?

 

[Doc Al]

2. If v=-gt is the velocity of the object relative to the ground, the velocity for an observer inside the elevator must be v'=-gt-V.

Presumably the object is released from rest relative to the elevator? If so, then v = -gt is its velocity relative to the elevator and -gt+V is its velocity relative to the ground.

 

[henry_m]

You've made two errors:
1) Relative to the fixed frame, the floor of the lift is moving downwards. When the ball bounces, it will not just reverse its velocity. What do you expect to happen if the lift is falling very quickly, and the ball is only just catching the floor up?
2) Bear in mind that by the time the ball has bounced, the elevator will have moved down so you must compensate for this in calculating the new height from the lift floor.

 

[cjl]

Henry: read the question again. The elevator is moving upwards, not downwards.

llandau: I agree with Doc Al. I would assume that the object began at rest relative to the elevator.

 

[henry_m]

Henry: read the question again. The elevator is moving upwards, not downwards.

Oops, sorry misread that. Or: I took at as moving downwards with speed -V .Either way, the point still stands. What if the elevator is moving upwards at enormous speed, and the ball is going really slowly when it hits the floor? You certainly wouldn't expect it to just reverse its direction and maintain its speed, since it'd have to fall out the bottom of the lift to do that. It has to reverse its direction and maintain its speed relative to the floor of the lift.

And for the record, I also agree with Doc Al and cjl.