**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.

**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.

- .