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Chapter 6: Problem 19

A space vehicle is traveling at \(3860 \mathrm{~km} / \mathrm{h}\) with respect to the Earth when the exhausted rocket motor is disengaged and sent backward with a speed of \(125 \mathrm{~km} / \mathrm{h}\) with respect to the command module. The mass of the motor is four times the mass of the module. What is the speed of the command module after the separation?

Short Answer

Expert verified
The speed of the command module after the separation can be calculated using the principle of conservation of momentum. Substitute the known values into the equation from step 4 to find the result.

Step by step solution

01

Understanding the problem

We know that the initial speed of the space vehicle (which includes both the rocket and command module) is 3860 km/hr. The mass of the rocket is four times the mass of the command module. The rocket motor is disengaged and is sent backward with a speed of 125 km/hr with respect to the command module.
02

Define and Assign Variables

Let's denote the mass of the command module as \(m\), hence the rocket's mass is \(4m\). The initial velocity of the system \(V_{i} = 3860\ km/hr\), the rocket's speed after disconnection \(V_{r} = -125\ km/hr\) (negative indicates opposite direction) and we need to find the speed of the command module \(V_{cm}\) after separation.
03

Conservation of Momentum

According to the principle of conservation of momentum, the total momentum before separation equals the total momentum after separation: \[(m+4m) \cdot V_{i} = m \cdot V_{cm} + 4m \cdot V_{r}\]
04

Solve for the command module speed

Rearrange the equation in step 3 to solve for \(V_{cm}\): \[V_{cm} = \frac{(m+4m) \cdot V_{i} - 4m \cdot V_{r}}{m}\] Substitute the known values into the equation we get the speed of the command module after separation.

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Most popular questions from this chapter

A \(75.2\) -kg man is riding on a \(38.6-\mathrm{kg}\) cart traveling at a speed of \(2.33 \mathrm{~m} / \mathrm{s} .\) He jumps off in such a way as to land on the ground with zero horizontal speed. Find the resulting change in the speed of the cart.

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