A car is traveling in the +x direction with a speed v when the driver notices a

Question

A car is traveling in the +x direction with a speed v when the driver notices a car that is stopped in front of him a distance d . in order to avoid a collision , he immediately applies the brakes resulting in an acceleration of -a1. at the same time ,the parcked car starts to acclelerate in the +x direction with an acceleration of a2. determine the distance d in terms of v, a1 and a2 if the cars are to just barely avoid a collision .

PLEASE DONT ANSWER THE QUSTION UNLESS YOU ARE 100% SURE :)

Explanation / Answer

Use the equation for the position of a body moving with uniform acceleration … … x = x0 + v0 t + ½ a t ² …that means, for the car at the back … … x1 = x10 + v10 t1 + ½ a1 t1 ² = 0 + v t1 + ½ (-a) t1 ² = v t1 - ½ a t1 ² … For the car at the front … x2 = x20 + v20 t2 + ½ a2 t2 ² = d + 0 + ½ (+A ) t2 ² … … x2 = d + ½ A t2 ² … about to collide … --> … x1 = x2 … with … t1 = t2 = t … That is … v t - ½ a t ² = d + ½ A t ² … --> … d = v t - ½ [ a + A ] t ² … Just barely avoid collision … --> … for the car at the back, the final velocity must be … 0 = v – a t … --> … t = v / a … --> … d = v ( v / a ) - ½ [ a + A ] ( v / a ) ² … … d = ( v ² / a ) - ½ [ ( v ² / a ) + A ( v ² / a ² ) ] …… = ½ [ ( v ² / a ) - A ( v ² / a ² ) ] …… = [ a – A ] v ² / ( 2 a ² )

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