Question

the driver of a car slams on the brakes, causing the car to slow down at a rate of 16.0ft/s² as the car skids 372ft to a stop. how long does the car take to stop? what was the cars initial speed?

Explanation:

Step1: Use the kinematic equations

We know the kinematic equations \(v = v_0+at\) and \(x = v_0t+\frac{1}{2}at^{2}\). Since the car stops, \(v = 0\). From \(v = v_0+at\), we have \(v_0=-at\). Substitute \(v_0=-at\) into \(x = v_0t+\frac{1}{2}at^{2}\).

Step2: Solve for \(t\)

Substituting \(v_0=-at\) into \(x = v_0t+\frac{1}{2}at^{2}\), we get \(x=-at^{2}+\frac{1}{2}at^{2}=-\frac{1}{2}at^{2}\). Given \(a=- 16.0\mathrm{ft/s^{2}}\) and \(x = 372\mathrm{ft}\). Then \(372=-\frac{1}{2}\times(-16)t^{2}\), \(372 = 8t^{2}\), \(t^{2}=\frac{372}{8}=46.5\), \(t=\sqrt{46.5}\approx6.82\mathrm{s}\)

Step3: Solve for \(v_0\)

Since \(v_0=-at\) and \(a=-16\mathrm{ft/s^{2}}\), \(t = 6.82\mathrm{s}\), then \(v_0=-(-16)\times6.82 = 109.12\mathrm{ft/s}\)

Answer:

The car takes approximately \(6.82\mathrm{s}\) to stop and the initial speed is approximately \(109.12\mathrm{ft/s}\)