Question

A body with zero initial velocity moves down an inclined plane from a height h and then ascends along the same plane with an initial velocity such that it stops at the same height h. In which case is the time of motion longer?

Answer 1

Let us first assume that there is no friction. Then according to the energy conservation law, the velocity v of the body sliding down the inclined plane from the height h at the foot is equal to the velocity which must be imparted to the body for its ascent to the same height h. Since for a body moving up and down an inclined plane the magnitude of acceleration is the same, the time of ascent will be equal to the time of descent.

If, however, friction is taken into consideration, the velocity v₁ of the body at the end of the descent is smaller than the velocity v (due to the work done against friction), while the velocity v₂ that has to be imparted to the body for raising it along the inclined plane is larger than v for the same reason. Since the descent and ascent occur with constant (although different) accelerations, and the traversed path is the same, the time t₁ of descent and the time t₂ of ascent can be found from the formulas

where s is the distance covered along the inclined plane. Since the inequality v₁ < v₂ is satisfied, it follows that t₁ > t₂. Thus, in the presence of sliding friction, the time of descent from the height h is longer than the time of ascent to the same height.

While solving the problem, we neglected an air drag. Nevertheless, it can easily be shown that if an air drag is present in addition to the force of gravity and the normal reaction of the inclined plane, the time of descent is always longer than the time of ascent irrespective of the type of this force. Indeed, if in the process of ascent the body attains an intermediate height h', its velocity v' at this point, required to reach the height h in the presence of drag, must be higher than the velocity in the absence of drag since a fraction of the kinetic energy will be transformed into heat during the subsequent ascent. The body sliding down from the height h and reaching the height h' will have (due to the work done by the drag force) a velocity v" which is lower than the velocity of the body moving down without a dray. Thus, while passing by the same point on the inclined plane, the ascending body has a higher velocity than the descending body. For this reason, the ascending body will cover a small distance in the vicinity of point h' in a shorter time than the descending body. Dividing the entire path into small regions, we see that each region will be traversed by the ascending body in a shorter time than by the descending body. Consequently, the total time of ascent will be shorter than the time of descent.