Question

Far in space, where gravity is negligible, a 440 kg rocket traveling at 75 m/s fires itsengines. Figure P9.26 shows the thrust force as a function of time,with the horizontal axis in 11 sincrements. The mass lost by the rocket during these 33 s is negligible.

Figure P9.26

(a) What impulse does the engine impart to therocket?
N·s
(b) At what time does the rocket reach its maximum speed?
s
What is the maximum speed?
m/s

Answer 1

Concepts and reason

The concepts used to solve this problem are impulse momentum theorem.

Find the expression for impulse from impulse-momentum theorem considering impulse caused by a force is equal to the area under the force time graph.

Use the impulse momentum theorem to find the time does the rocket reaches its maximum speed.

Finally find the maximum speed by using impulse momentum relation.

Fundamentals

The expression for impulse is,

Here, is the impulse, is the force, is the initial time, and is the final time.

The impulse is equal to the area under the curve between and .

Expression for area of the triangle is,

Here, A is Area of the triangle, is the base and is the height.

The impulse caused by a force is equal to the area under the force time graph.

Expression for impulse is,

…… (1)

From impulse momentum theorem,

The change in momentum of an object equals the impulse applied to it.

…… (2)

Momentum is the product of velocity and mass of an object.

Here, is the momentum, is the mass, and is the velocity.

The initial momentum can be written as,

Here, is the initial momentum, is the initial velocity.

And the final momentum can be written as,

Here, is the final momentum, is the final velocity.

Substituting and in (2).

By rearranging the above equation,

The expression for the final momentum is,

…… (3)

And also the expression for maximum speed is,

…… (4)

(a)

The expression for the impulse does the engine impart to the rocket is,

Substitute for and for .

(b.1)

There is no force acting against the direction of motion of the rocket in the space. So, the only force acting on it is the thrust force of engine on it.

From the given force vs time graph.

The rocket accelerate from t=0 s to t=33 s. That is till t=33 s the rocket accelerates starting from t=0 s. If the rocket accelerates then the speed of the rocket keep on increasing.

So, the maximum speed the rocket reaches it at t=33 s.

The rocket reaches its maximum speed at , because it keeps accelerating until the booster stop pushing.

(b.2)

The expression for the final momentum is,

From the above the expression for maximum speed is,

Substitute for , for , and for .

Ans: Part a

The impulse does the engine impart to the rocket is .