Problem $1(20\%)$

A deer crossing a highway with a speed limit of $45mph$ led to the sudden braking of the traffic

and caused an accident between two vehicles. The back vehicle hit the rear bumper of the front

vehicle and caused severe damage to both vehicles. An accident investigation through structural

analysis indicates that the impact speed was $40mph$ when the accident occurred. Data from the

surveillance camera showed that both vehicles were going at about the same speed and the

clearance between them is $60$ feet.

Estimate how fast were the two vehicles going before the crash.

What would be the smallest safety clearance between the two vehicles at the estimated

speed above, if the front vehicle ran into a stopped truck at night? $($Assume a reaction

time of $2\mathrm{.}5sec,$ car length $=18ft,$ and a coefficient of friction of $0\mathrm{.}65.$