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# Dangerous Driver?

##### Age 16 to 18Challenge Level

We had two very interesting solutions, which were beautifully presented Word documents -- click below to read them

Michael from Ecclesbourne
Michael believes that the penalty could not reasonably be rejected as the situation stands -- more information will be required about the specific car in question because, quite rightly, acceleration will not be constant in a real situation.

Henry, from Elizabeth College
Henry carefully converted all of the units and performed a calculation based on constant acceleration and various equations of mechanics. Based on these assumptions, he concludes that the car could be going as fast as 41m/s. This is greater that the speed that the camera recorded and that the case should not be dismissed on mathematical grounds.

Steve notes
In reality I wondered if constant power produced by the car might be a more solid starting point for a calculation. In principle this could be inferred from the solid data point of acceleration from 0 to 96 km/h in 10.5 seconds. A big unknown would be the retarding effect of wind resistance. Another big unknown would be the road configuration. Is it curved, straight, flat or up/downhill?

Patrick sent us his solution, which considers how air resistance might affect the problem.

He used the formula for air resistance, $F_{air} =\frac 12 A C_d D v^2$, where $A$ is the cross-sectional area of the car, $C_d$ is a constant saying how resistive air is and $D$ is the density of the air.

He also used the chain rule to write acceleration as:
$\frac{dv}{dt} = \frac{dx}{dt} \frac{dv}{dx} = v \frac{dv}{dx}$