1.4 Projectile motion types

So far, we have limited our discussion to the classic mode of a projectile motion, where points of projection and return are on the same horizontal plane. This situation, however, may be altered. The projection level may be at an elevation with respect to the plane where projectile returns or the projection level may be at a lower level with respect to the plane where projectile returns. The two situations are illustrated in the figures below.

The two variants are basically the same parabolic motion. These motion types are inherently similar to the one where points of projection and return are on same level. If we look closely, then we find that the motion of the projectile from an elevation and from a lower level are either an extension or a curtailment of normal parabolic motion.

The projection on an incline (where point of return is on higher level) is a shortened projectile motion as if projectile has been stopped before returning to the normal point of return. This motion is also visualized as if the projectile is thrown over an incline or a wedge as shown in the figure. Again, there are two possibilities : (i) the projectile can be thrown up the incline or (ii) the projectile can be thrown down the incline. For the sake of convenience and better organization, we shall study projectile motion on an incline in a separate module. In this module, we shall restrict ourselves to the first case in which projectile is projected from an elevation.

Projection from a higher level

The projection of a projectile from a higher point results in a slightly different parabolic trajectory. We can visually recognize certain perceptible differences from the normal case as listed here :

• The upward trajectory is smaller than downward trajectory.
• Time of ascent is smaller than the time of descent.
• The speed of projection is not equal to speed of return on the ground.
• The velocity of return is more aligned to vertical as the motion progresses.

It is evident that the expressions derived earlier for time of flight (T), maximum height (H) and range (R) are not valid in the changed scenario. But the basic consideration of the analysis is necessarily same. The important aspect of projectile motion that motions in two mutually perpendicular directions are independent of each other, still, holds. Further, the nature of motion in two directions is same as before : the motion in vertical direction is accelerated due to gravity, whereas motion in horizontal direction has no acceleration.

Now, there are two important variations of this projectile motion, when projected from an elevated level. The projectile may either be projected at certain angle (up or down) with the horizontal or it may be projected in the horizontal direction.