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Chapter 4: Problem 61

A boy coasts down a hill on a sled, reaching a level surface at the bottom with a speed of \(7.0 \mathrm{~m} / \mathrm{s}\). If the coefficient of friction between the sled's runners and the snow is \(0.050\) and the boy and sled together weigh \(600 \mathrm{~N}\), how far does the sled travel on the level surface before coming to rest?

Short Answer

Expert verified
The solution steps reveal that the critical aspect of this problem is utilizing the work-energy theorem and relating the work done by the force of friction with the kinetic energy of the system. By setting these equal, one can solve for the unknown distance the sled travels before coming to rest.

Step by step solution

01

Calculating the Initial Kinetic Energy

First, calculate the initial kinetic energy (K.E.) of the sled using the formula of kinetic energy \(K.E. = \frac{1}{2}m v^2\), where \(v = 7.0 m/s\) is the initial speed. Here, the mass \(m\) of the boy and sled together can be calculated using the formula \(m =\frac{F}{g}\), where \(F = 600 N\) is the force (also read as weight of the boy and the sled) and \(g = 9.81 m/s^2\) is acceleration due to gravity.
02

Calculating the Work Done by Friction

Next, calculate the work done by the force of friction. According to the work-energy theorem, this work is equal to the decrease in kinetic energy, which is simply the initial kinetic energy as calculated above, as the final kinetic energy is zero when the sled comes to rest. The work done by friction can be calculated as \(W = F_f d\), where \(F_f\) is the friction force and \(d\) is the distance to be calculated. The friction force can be calculated as \(F_f = \mu F\), where \(\mu = 0.050\) is the coefficient of friction and \(F\) is the weight of the boy and sled (600 N).
03

Solving for the Distance

Set the work done by friction equal to the initial kinetic energy and solve the equation for distance \(d\). This will give you the distance the sled travels before coming to rest.

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