Two trains, A A\,A and BBB, are moving in the same direction along the same straight horizontal track.
Train A A\,A moves with constant speed 20 m s−1^{-1}−1.
At a certain instant train B B\,B is 500 metres behind train A A\,A and is moving with speed 35 m s−1^{-1}−1.
At that instant the driver of B B\,B applies the brakes, producing a constant deceleration of magnitude d d\,d m s−2^{-2}−2.
Write down expressions for the distances travelled by trains A A\,A and B B\,B in the first t t\,t seconds after the brakes are applied.
Hence find the least value of d d\,d for which B B\,B does not overtake AAA.
State one modelling assumption you have made.