Problem 33.

A rocket is fired vertically upwards with initial velocity 80 m/s at the ground level. Its engines then fire and it is accelerated at until it reaches an altitude of 1000 m. At that point the engines fail and the rocket goes into free-fall. Disregard air resistance.

(a)  How long was the rocket above the ground?

(b)  What is the maximum altitude?

(c) What is the velocity just before it collides with the ground?

Solution:

The first part of the motion is the motion with constant acceleration at . The initial velocity for this motion is 80 m/s. Then we can write the equation, which describe the dependence of height of the rocket on time:

From this equation we can find the time when the rocket reach the height 1000 m = h:

The solution of this equation is 10 s. So after 10 seconds the engine fails. The velocity at this moment of time is

After this moment of time we have free fall motion – there is only one force acting on the object (it is gravitational force) – this force provide free fall acceleration.

The initial velocity is 120 m/s pointing upward. The acceleration is pointing downward. The initial height of the rocket is 1000 m. Then the equations which describe this motion are the following:

To find the maximum height of the rocket we can use the last equation. The velocity at the maximum height is 0. Then

This is the answer to part (b).

To find the time when the rocket hits the ground we need to use the first equation:

When the rocket hits the ground h=0. Then

From this equation we can find time: 31 s.

Then we can find the time when the rocket is in the air: it is the sum of the time when it reaches 1000 m and the time when it hits the ground:

This is the answer to part (a).

To find the speed of the rocket when it hits the ground we need to use the last equation:

When the missile hits the ground h=0. Then

This is the answer to part (c).