Let  denote the rocket's position, velocity, and acceleration vectors at time
 denote the rocket's position, velocity, and acceleration vectors at time  .
.
We're given its initial position

and velocity

Immediately after launch, the rocket is subject to gravity, so its acceleration is

where  .
.
a. We can obtain the velocity and position vectors by respectively integrating the acceleration and velocity functions. By the fundamental theorem of calculus,


(the integral of 0 is a constant, but it ultimately doesn't matter in this case)

and



b. The rocket stays in the air for as long as it takes until  , where
, where  is the
 is the  -component of the position vector.
-component of the position vector.

The range of the rocket is the distance between the rocket's final position and the origin (0, 0, 0):

c. The rocket reaches its maximum height when its vertical velocity (the  -component) is 0, at which point we have
-component) is 0, at which point we have

