A single–lane road is full of cars which move at the speed $υ = 90 k m / h$ . The average distance between neighbouring cars is such that a standing observer would measure a time lapse $τ = 2 s$ from the head of the first car until the head of the second car. One car has to stop due to malfunctioning and a queue of standing cars starts forming behind it. Find the speed $u$ with which the queue length is growing if the average distance between the centres of neighbouring cars in the queue is $l = 6 m$ ?

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answer is 3.4.

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Detailed Solution

Consider a reference frame moving with speed $u$ opposite to the direction of the cars. Cars at the end will be moving with speed $υ + u$ , and the distance between them will be $υ τ$ . Cars at the front will be moving at speed $u$ (because they are stopped) and have a distance $l$ between them.
Now we take ratios of speed to length and equate them to get
$\frac{υ τ}{υ + u} = \frac{l}{u}$