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Chapter 6: Problem 16

A 195-lb man standing on a surface of negligible friction kicks forward a \(0.158\) -lb stone lying at his feet so that it acquires a speed of \(12.7 \mathrm{ft} / \mathrm{s}\). What velocity does the man acquire as a result?

Short Answer

Expert verified
The velocity acquired by the man as a result of the kick is approximately -0.0103 ft/s, in the direction opposite to the stone's motion.

Step by step solution

01

Set Up the Momentum Conservation Equation

According to the principle of conservation of momentum, the total momentum of a system before and after an event (in this case the kick) must remain the same. Therefore, the initial momentum (before the kick) must be equal to the final momentum (after the kick). Considering only the horizontal direction (as all movements are horizontal), the equation of conservation of momentum can be written as, \(m_1 \cdot v_1 + m_2 \cdot v_2 = m_1 \cdot v_1' + m_2 \cdot v_2'\) where \(m_1\) and \(m_2\) are the masses of the man and the stone respectively, \(v_1\) and \(v_2\) are their initial velocities (which are zero before the kick), and \(v_1'\) and \(v_2'\) are their final velocities.
02

Substitute Known Values

Now we substitute the known values into the equation. According to the question, the man's mass \(m_1\) is 195 lbs, the stone's mass \(m_2\) is 0.158 lbs, the stone's final velocity \(v_2'\) is 12.7 ft/s, and \(v_1\) and \(v_2\) are zero (since both the man and the stone were initially at rest). The man's final velocity \(v_1'\) is the unknown we want to find. So we can put these values into the equation. After substitution, the equation will be 0 + 0 = 195 lbs \cdot v_1' + 0.158 lbs \cdot 12.7 ft/s.
03

Solve for the Unknown

From step 2 we need to solve the equation to get the final velocity of the man \(v_1'\). Simplifying the equation, we get \(v_1'\) = - (0.158 lbs \cdot 12.7 ft/s) / 195 lbs. The negative sign indicates that the direction of man's movement is opposite to that of the stone.

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