# In terms of being on an elevator, velocity, acceleration and forces, how are they related?

Before you answer, read this: Say you are on an elevator, and for some reason, when in it and you jump, you have a different velocity from the elevator, say the elevator is at 15 MPH and you are jumping at 5 MPH. The elevator catches up faster than it does when you and the elevator are both at the same speed, right? Also, when you jump on the elevator while you and it are at the same velocity, and you land as the elevator is accelerating upward, and additionally in the hypothetical case where the elevator is going faster than you when you jump as well; when you land on the faster, whether it is accelerating upward or otherwise going faster than you when you jump, would you feel more force when you land in either case than if the elevator were to go slower than you or accelerate downward when jumping? Why?

The easiest way to understand this is using relative motion and pseudo forces. Assume at an instant you have a velocity of $$u$$, and the lift has a velocity of $$v$$, analyzing it in the lift frame, your velocity will be $$u-v$$. If the lift is accelerating with acceleration $$a$$, you'll feel a pseudo force equal to $$ma$$ (where $$m$$ is your mass) in the direction opposite to the acceleration. This means that if the lift is accelerating upwards, your downward acceleration in the lift frame will be $$g+a$$, and if its accelerating downwards, it will be $$g_a$$. Using these adjustments to velocity and acceleration, you can now analyze this like any normal stationary frame.
Note that when you land, the force you think you feel is actually the impulse. Assuming you don't bounce back up, that impulse will be $$mv_{rel}$$, where $$v_{rel}$$ is your velocity with respect to the lift at that time. So the acceleration of the lift doesn't affect the impulse you feel, only the velocity does. If the lift is moving up, you'll feel a greater impulse, and if its moving down, you'll feel a lesser impulse, irrespective of the magnitude and direction of the acceleration.