Question

A car moving with speed v on a straight road can be stopped within distance d on applying brakes. If the same car is moving with speed 3v and brakes provide half retardation, then the car will stop after travelling distance.

Answer 1

Hint: To find the distance at which the car stops, first we need to find the acceleration of the car after the brakes are applied. After finding the acceleration we will find the distance required to stop the car if the acceleration or retardation is halved. The formula used is:
\[{{V}^{2}}={{U}^{2}}+2aS\]
where V is the final velocity, U is the initial velocity, a is the acceleration and S is the distance stopped.

Complete step by step answer:
The speed with which the car is moving is given as velocity v.
Now, the car has stopped due to the application of the brakes making the final velocity zero.
Hence, the distance to stoppage is d and the initial velocity is v making the acceleration of the car till stoppage as a, and to find the value of "a", we use the formula of \[{{V}^{2}}={{U}^{2}}+2aS\].
After finding the acceleration, we place the value of the acceleration in the second case or the case when the car comes to stop and as given in the question, the car's acceleration is reduced to half hence, the acceleration when the car retards after braking is \[\dfrac{a}{2}\]. Now the velocity with which the car moves after retardation is \[3v\] and acceleration is \[a=\dfrac{{{v}^{2}}}{4d}\]. Placing all the values in the distance formula we get the value of the distance covered after braking is:
\[{{V}^{2}}={{U}^{2}}+2aS\]
Placing the value of v as \[3v\] and a as \[\dfrac{{{v}^{2}}}{4d}\], we get the distance as:
\[S=\dfrac{{{(3v)}^{2}}}{2\left( \dfrac{{{v}^{2}}}{4d} \right)}\]
\[S\text{ }=\text{ }18d\]
Therefore, after braking the car will stop at a distance of \[18d\].

Note:
The term retardation means backward or negative acceleration where the acceleration is used during braking purposes, so the retardation is halved meaning the acceleration becomes half of that of the acceleration when the car is moving forward.