Understanding Falling with Horizontal Velocity

[Figaro]

In the book I'm reading, there is a discussion about how a person inside a "box" falling in a gravitation field would see himself as compared to someone on the ground. It is as follows

"While the guy is asleep, put him in a spacious box elaborately furnished inside to look exactly like his living
room. We then drop the box from a high-flying airplane. When our friend wakes up, he thinks that he is in his living room. Curiously, though, he feels that he is floating. To an observer on the ground, our friend and his living room are hurtling toward a crunching rendezvous with the ground. Our friend, however, is blissfully unaware of the impending disaster. Since he is accelerating downward at the same rate as the box and all the objects contained inside, he feels that he is not moving downward at all relative to his surroundings. A slight spring in his step and he finds himself drifting toward the ceiling. He feels that he is floating. But this action is interpreted by the ground observer quite differently: our friend, by stepping on the floor, has at the same time decreased slightly his downward velocity and increased slightly the box’s downward velocity. He thinks he is floating upward but in reality his downward plunge is accelerating at the same rate as before."

My questions here are:
1) Where is the floor and ceiling according to the image? It seems unclear as to where it is although given the position of the window, the ceiling should be the top face.
2) My confusion is somehow originating from question 1, if the floor is on the "side" say, left face of the room (since his feet is on that face although it's weird to have a ceiling with windows), by stepping on that face, the outside observer would view him having a horizontal velocity, but horizontal velocity doesn't affect the downward acceleration. So I think what he had done was that he stepped on the bottom face (with the carpet?) and having upward velocity, but this motion was perceived by the ground observer as the guy decreasing his downward velocity and the box gaining downward velocity.
3) What is the statement "accelerating at the same rate as before" pertaining? If he stepped on the floor, he gained upward velocity so the acceleration should decrease right? Or is it the gravitational acceleration in which regardless of what you are doing, it is pulling you at the same rate?

 

[Ibix]

I agree the diagram is confusing. I think the guy is supposed to be "floating in mid-air" on his back. Presumably he was lying on his back after we knocked him out and floated upwards when he pushed off the floor trying to stand up.

As he kicked off the floor he did decelerate briefly, yes. Once he's no longer touching the floor, though, he is again accelerating at one gravity as seen by the guy on the ground, just traveling slightly slower than the boxuntil he bumps into the roof.

 

[Figaro]

I agree the diagram is confusing. I think the guy is supposed to be "floating in mid-air" on his back. Presumably he was lying on his back after we knocked him out and floated upwards when he pushed off the floor trying to stand up.

As he kicked off the floor he did decelerate briefly, yes. Once he's no longer touching the floor, though, he is again accelerating at one gravity as seen by the guy on the ground, just traveling slightly slower than the boxuntil he bumps into the roof.

Thanks for the clarification but somehow the last sentence is confusing "He thinks he is floating upward but in reality his downward plunge is accelerating at the same rate as before." First, the process of him floating upward is only known to him since he doesn't know that he is indeed falling, so it doesn't have anything to do with him KNOWING that his acceleration is the same as before. Only the outside observer can tell that even though he "floated" upward for a while, he is still falling at the same rate, and it is kinda redundant to say the last statement knowing that the ground observer already knows that the guy inside is falling, so he should have the same acceleration otherwise.

 

[Ibix]

Where is this coming from? It's not very clear, I think.

Neither view is right or wrong or "in reality". The guy in the box is floating upwards in zero g ascfar as he is concerned. As far as the person on the ground is concerned both he and the room are accelerating at 9.81ms ^-2, but the room is always moving a couple of centimeters per second faster than the man.

 

[Figaro]

Where is this coming from? It's not very clear, I think.

Neither view is right or wrong or "in reality". The guy in the box is floating upwards in zero g ascfar as he is concerned. As far as the person on the ground is concerned both he and the room are accelerating at 9.81ms ^-2, but the room is always moving a couple of centimeters per second faster than the man.

The room is always moving a couple of centimeters per second faster than the man during the process when the man stepped on the floor but eventually acceleration will counteract the upward motion and he will fall at the same rate again right? Also, given that the room is big or his upward velocity is small enough so that he will not reach the ceiling, he doesn't need to bump into the ceiling before he starts falling at the same rate (as you've said in post #2) again right?

 

[PeroK]

The room is always moving a couple of centimeters per second faster than the man during the process when the man stepped on the floor but eventually acceleration will counteract the upward motion and he will fall at the same rate again right? Also, given that the room is big or his upward velocity is small enough so that he will not reach the ceiling, he doesn't need to bump into the ceiling before he starts falling at the same rate (as you've said in post #2) again right?

The whole point of this thought experiment is that, while the man is in his room, gravity is doing precisely nothing to him. To him, there is no gravity. Nothing is accelerating.

 

[PeterDonis]

eventually acceleration will counteract the upward motion and he will fall at the same rate again right?

Not relative to the room, no. Relative to the room, the man is not accelerating at all. That's the point.

given that the room is big or his upward velocity is small enough so that he will not reach the ceiling, he doesn't need to bump into the ceiling before he starts falling at the same rate (as you've said in post #2) again right?

No. Until the man bumps into the ceiling, he will continue to rise at a constant velocity relative to the room. (Unless of course the whole thing hits the ground before that happens.)

 

[A.T.]

The room is always moving a couple of centimeters per second faster than the man during the process when the man stepped on the floor but eventually acceleration will counteract the upward motion and he will fall at the same rate again right?

No. Since both have the same acceleration, the man cannot catch up with the room, until he reaches the ceiling. The ceiling will then apply a downwards force to him, increasing his acceleration temporally, so he will reach the same vertical speed as the room (or higher if he bounces from the ceiling).

 

[Ibix]

Say that, according to the outside observer, the room has accelerated to 100m/s when the man pushes off. He slows down slightly and is now moving at 99.98m/s, while the room speeds up slightly and is now moving at 100.001m/s. They both continue to accelerate at 9.8m/s². So at a time $t$ seconds after the man pushes off the room is falling at $(100.001+9.8t)$m/s and the man is falling at $(99.98+9.8t)$m/s. The relative velocity between the two is $(100.001+9.8t-(99.98+9.8t))=0.021$m/s, and is independent of time.

The man and the room are accelerating at the same rate. They are not moving at the same speed. This is true in all frames.

 

[Figaro]

Say that, according to the outside observer, the room has accelerated to 100m/s when the man pushes off. He slows down slightly and is now moving at 99.98m/s, while the room speeds up slightly and is now moving at 100.001m/s. They both continue to accelerate at 9.8m/s². So at a time $t$ seconds after the man pushes off the room is falling at $(100.001+9.8t)$m/s and the man is falling at $(99.98+9.8t)$m/s. The relative velocity between the two is $(100.001+9.8t-(99.98+9.8t))=0.021$m/s, and is independent of time.

The man and the room are accelerating at the same rate. They are not moving at the same speed. This is true in all frames.

This example really cleared up the confusion. They are not moving at the same speed, so the room will eventually hit the guy. Though in the whole process, the guy is falling at the same rate as the room. Thanks for the clarification!