Question

A truck ran into a tree trunk that fell across the road. The driver claims that he was going the speed limit, 45 mph (20 m/s), and as soon as the tree fell, he slammed on his brakes, but there was no way for him to stop in time. Accident investigators found that the car left skid marks 43.4 m long. You know that, on dry pavement, a car of that size with that type of tires will decelerate at 6.5 m/s2 when skidding. Estimate his actual speed.

Answer 1

To get the estimate of the actual speed, let us consider the case of initial speed, such that the truck stops just before the tree. The equations of motion are:

$v(t) = v_0 - a t$, $S(t) = v_0 t - \frac{a t^2}{2}$.

The time it takes to stop is found from $v(t') = v_0 - a t' = 0$, from where $t' = \frac{v_0}{a}$.

The displacement in that time is $S(t') = \frac{v_0^2}{a} - \frac{v_0^2}{2 a} = \frac{v_0^2}{2 a}$. Hence, given the length of skid marks $S = 43.4 m$, from the last equation the initial speed it would take to completely stop in front of the tree is $v_0 = \sqrt{2 S a} \approx 23.75 \frac{m}{s}$. Since the truck ran into the tree, the speed must have been higher than $23.75 \frac{m}{s}$.