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Chapter 7: Problem 9

An \(84.4\) -kg man is standing at the rear of a \(425-\mathrm{kg}\) iceboat that is moving at \(4.16 \mathrm{~m} / \mathrm{s}\) across ice that may be considered to be frictionless. He decides to walk to the front of the \(18.2-\mathrm{m}\) -long boat and does so at a speed of \(2.08 \mathrm{~m} / \mathrm{s}\) with respect to the boat. How far does the boat move across the ice while he is walking?

Short Answer

Expert verified
The calculation will give us the exact distance the boat moves while the man is walking, by using the conservation of momentum principle and solving for the boat's velocity change due to man's velocity changed.

Step by step solution

01

The initial and final momentums

Firstly, calculate the initial momentum of the system (man and boat) before the man starts walking: \( P_{initial} = (mass_{man} + mass_{boat}) * velocity_{boat} = (84.4 kg + 425 kg) * 4.16 m/s = 2118.64 kg.m/s \). Then, calculate the final momentum of the system after the man has reached the front of the boat, which will still equal the initial momentum.
02

The changed velocities

The final momentum of the system will be the sum of momentum of the man and that of the boat: \( mass_{man} * velocity_{man} + mass_{boat} * velocity_{boat(new)} = 2118.64 kg.m/s \). Here, \( velocity_{man} = 2.08 m/s \) is the velocity of man relative to the boat. The \( velocity_{boat(new)} \) is what we need to find in order to calculate how far the boat moves during the period when the man walks. Solve this equation to find \( velocity_{boat(new)} \).
03

The travelled distance

Calculate how long it takes the man to walk to the front of the boat: \( time = distance / speed = 18.2 m / 2.08 m/s = 8.75 s \). Then, multiply this time by \( velocity_{boat(new)} \) to get the distance the boat moves while the man is walking.

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