How is it true that your feet press harder on the floor of an elevator which is rising at constant speed than when the elevator is going down at the same speed?

Answer 1

Inertia plus Gravity
Inertia is the tendency of a stationary object to remain stationary and of a moving object to remain in motion. The elevator is exerting pressure against your body to get it moving up and gravity is pushing you down so you feel pressure. When going down the elevator is simply falling away from you at a slightly faster pace than gravity can keep up so you feel less pressure. These are what are called fictitious forces. Fictitious forces are not real forces in the sense that they don't come from an outside object, but as a result of trying to keep up with a changing environment.

Alternate Answer

Assume that travel distances are realistic for elevators near earth's surface. If the elevator is at rest, or if it is moving at a constant velocity in any direction, it is what is called an inertial frame of reference. Experiments done in one inertial frame will yield the same results as in any other inertial frame. There is no gravitational lag. So at constant velocity, there is no perception of lightness or heaviness or "moving elevator". In short, there is no pressing harder or lighter to the floor. See discussion; there are important limiting considerations if you are moving great distances.

[Quoted from above: When going down the elevator is simply falling away from you at a slightly faster pace than gravity can keep up so you feel less pressure.] If this were true when going down at constant velocity, then you would continue to float higher and higher until you are stopped by the ceiling of the elevator. It would have to mean that the elevator is accelerating downward faster than an object in free-fall, which in fact you would be. So this cannot be happening under zero acceleration.