Problem

Consider a projectile fired at an initial velocity at firing angle of above the ground. Ignore air resistance or the effects of a rotational effects of the Earth.

Projectile motion.jpg

Part 1: Calculate the time it takes for the projectile to land at the same altitude.

Part 2: Calculate the range of the projectile in terms of , , and .

Part 3: Calculate the maximum altitude that the projectile is able to attain.

Solution

This problem must be analyzed in two directions.

Treat the initial point . There is no acceleration in the x-direction, and the acceleration in the x-direction is .

Part 1

Since the projectile lands at the same altitude, the displacement in the y-direction is 0.

This means the two times in which are and . Of course the second time is the landing time.

Part 2

.

Since , the range of the projectile is

.

Part 3

The top of the trajectory occurs at half the time of landing, so

.

Substitute this time to the displacement equation in the y-direction.

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