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#25 Car crash maths

Solution

A blue car travelling at 70mph and a yellow one at 100mph brake simultaneously when they see a fallen tree ahead. The blue one stops just in time. But the yellow one hits the tree at about 70mph, so option (d) is the best answer. The yellow car hasn’t even got down to the speed that the blue car was travelling at. Here is one way to calculate this:

The energy expended on braking is the force × the distance.

In this model, therefore, both cars expend the same amount of energy.

The blue car braked to a halt and has zero energy left.

Energy in motion (kinetic energy) is proportional to ½×mass×velocity², so think of both cars losing ½×m×70² of energy.

The yellow car started with ½×m×100².

70² is not quite half of 100², so the yellow car still has half of its energy, which means it is still travelling at a touch over 70mph when it hits the fallen tree.

The same result can be reached if you use the equations of constant acceleration. Our intuition is confused by the squaring of the velocities – this isn’t a linear relationship. This helps to explain why our guesses are often poor in such situations.

What’s more, this model assumes no driver reaction time and some simplification of braking forces. In fact, if reaction time is included, the yellow car will be travelling even faster when it hits the tree.

Quick Quiz #25

Answers

1 James Clerk Maxwell

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3 Polyps

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