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Chapter 7: Problem 23

A \(58-\) kg astronaut is in space, far from any objects that would exert a significant gravitational force on him. He would like to move toward his spaceship, but his jet pack is not functioning. He throws a 720 -g socket wrench with a velocity of \(5.0 \mathrm{m} / \mathrm{s}\) in a direction away from the ship. After \(0.50 \mathrm{s}\), he throws a 800 -g spanner in the same direction with a speed of \(8.0 \mathrm{m} / \mathrm{s} .\) After another $9.90 \mathrm{s}\(, he throws a mallet with a speed of \)6.0 \mathrm{m} / \mathrm{s}$ in the same direction. The mallet has a mass of \(1200 \mathrm{g}\) How fast is the astronaut moving after he throws the mallet?

Short Answer

Expert verified
Answer: The final velocity of the astronaut after throwing all the tools is approximately \(0.297 \, m/s\) towards the spaceship.

Step by step solution

01

Calculate the initial momentum of the astronaut-tool system.

Initially, the astronaut and all the tools are at rest. Therefore, the initial momentum of the system is 0.
02

Calculate the change in momentum during the first throw

The astronaut throws a 720 g socket wrench with a velocity of \(5.0m/s\) away from the ship. The change in momentum can be calculated using the formula \(\Delta p = m_{tool}v_{tool}\). Note that 720 g is equal to 0.72 kg. \(\Delta p_1 = (0.72 kg)(5.0 m/s) = 3.6 kg.m/s\)
03

Calculate the change in momentum during the second throw

The astronaut throws a 800 g spanner with a velocity of \(8.0m/s\) away from the ship. Similarly, we calculate the change in momentum using the same formula. Convert 800 g to 0.8 kg. \(\Delta p_2 = (0.8 kg)(8.0 m/s) = 6.4 kg.m/s\)
04

Calculate the change in momentum during the third throw

The astronaut throws a 1200 g mallet with a velocity of \(6.0m/s\) away from the ship. Again, calculate the change in momentum using the same formula. Convert 1200 g to 1.2 kg. \(\Delta p_3 = (1.2 kg)(6.0 m/s) = 7.2 kg.m/s\)
05

Calculate the total change in momentum

Add the change in momentum during each throw to find the total change in momentum. \(\Delta p_{total} = \Delta p_1 + \Delta p_2 + \Delta p_3 = 3.6 kg.m/s + 6.4 kg.m/s + 7.2 kg.m/s = 17.2 kg.m/s\)
06

Calculate the final velocity of the astronaut

Divide the total change in momentum by the astronaut's mass to find the final velocity of the astronaut. \(v_{astronaut} = \frac{\Delta p_{total}}{m_{astronaut}} = \frac{17.2 kg.m/s}{58 kg} = 0.2966 m/s\) The final velocity of the astronaut after throwing the mallet is approximately \(0.297 \, m/s\) towards the spaceship.

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