Problem 31.

A missile is launched into the air at an initial velocity of 80 m/s. It is moving with constant velocity until it reaches 1000m, when the engine fails. 

(a) How long does it take it to reach 1000m?

(b) How high does the missile go?

(c) How long does it take for it to fall back to the earth? 

(d) How long does it stay in the air?

(e) How fast is it going when it hits the ground?

Solution:

(a) Since initial we have the motion with constant velocity we can easily find the time of the motion of the missile till it reaches the height 1000 m. The time is given by the expression:

After this point 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 80 m/s pointing upward. The acceleration is pointing downward. The initial height of the missile is 1000 m. Then the equations which describe this motion are the following:

(b) To find the maximum height of the missile we can use the last equation. The velocity at the maximum height is 0. Then

(c) To find the time when the missile hits the ground we need to use the first equation:

When the missile hits the ground h=0. Then

From this equation we can find time: 24.6 s.

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

(e) To find the speed of the missile when it hits the ground we need to use the last equation:

When the missile hits the ground h=0. Then

Here we need to add the minus sign, which illustrate the fact that the direction of velocity is downward.