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Chapter 4: Problem 20

In a front-end collision, a \(1300-\mathrm{kg}\) car with shock-absorbing bumpers can withstand a maximum force of \(65 \mathrm{kN}\) before damage occurs. If the maximum speed for a nondamaging collision is \(10 \mathrm{km} / \mathrm{h},\) by how much must the bumper be able to move relative to the car?

Short Answer

Expert verified
To find the distance the bumper needs to move to avoid damage, substitute the values for kinetic energy and force into the equation for distance, \(d = \frac{KE}{F}\).

Step by step solution

01

Calculate the car's initial kinetic energy

First, it is necessary to convert the speed of the car from kilometers per hour (km/h) to meters per second (m/s). This can be done using the conversion factor \(1 \textrm{ km/h} = \frac{1}{3.6} \textrm{ m/s}\). Thus, \(v = 10 \textrm{ km/h} = \frac{10}{3.6} \textrm{ m/s}\). Then the kinetic energy (\(KE\)) of the car can be determined using the formula \(KE = \frac{1}{2} m v^2\), where \(m = 1300 \textrm{ kg}\) is the mass of the car and \(v\) is its speed.
02

Calculate the Work Done

Since there's no damage to the car, all the initial kinetic energy goes into performing work, which is used to stop the car. That is, the work done (\(W\)) equals to the initial kinetic energy, so \(W = KE\). We know that the work done is also equal to the force applied (\(F\)) multiplied by the distance traveled (\(d\)), i.e, \(W = Fd\).
03

Calculate the Distance

Substitute \(W = KE\) into the work done equation \(W = Fd\), it can be rewritten as \(Fd = KE\). Solve this equation for the distance \(d\) to get the equation \(d = \frac{KE}{F}\). Now we can substitute the previously calculated kinetic energy and force into this equation to find the distance. It's given that \(F = 65 \textrm{ kN} = 65,000 \textrm{ N}\), the force of impact.

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