Question

A car moving at a speed v is stopped by a retarding force F in a distance s. If the speed of the car is $3v,$ the force need to stop the car in a same direction will be
A. \[3F\]
B. \[6F\]
C. \[9F\]
D. \[12F\]

Answer 1

Hint: The retarding force is the resisting force which is against the direction of the object’s current velocity. For Example – For a car its retarding force is F. It is the force which causes the acceleration of an object to be negative. Here use the equation related to the initial velocity and the final velocity.

Complete step by step answer:
Let us assume that the mass of the car be $m$
Distance travelled be $s = x$
The final velocity of the car is $v = 0$
Now, by using the equation of motion –
${v^2} - {u^2} = 2as$
Place the values of velocities and acceleration in form of force ($a = \dfrac{{ - F}}{m}$) it is negative since the force is retarding in the above equation –
$0 - {v^2} = 2\left( {\dfrac{{ - F}}{m}} \right)x$
Negative signs on both the sides of the equations cancel each other and both the sides of the equation become positive.
${v^2} = 2\left( {\dfrac{F}{m}} \right)x{\text{ }}......{\text{(a)}}$
Now, given that if the velocity of the car is made three times
${(3v)^2} = 2\left( {\dfrac{{F'}}{m}} \right)x$
Simplify the above equation –
$9{v^2} = 2\left( {\dfrac{{F'}}{m}} \right)x{\text{ }}......{\text{(b)}}$
 To compare force F’ and F, multiply equation (a) by $9$
$9{v^2} = 9 \times 2\left( {\dfrac{F}{m}} \right)x{\text{ }}......{\text{(c)}}$
Now comparing equations (b) and (c)
$F' = 9F$
Therefore, the required answer is - the force need to stop the car in a same direction will be $9F$

So, the correct answer is “Option C”.

Note:
The equations of the motions are used to know the behaviour of the certain physical parameters in the system such as distance, velocity and acceleration. Always remember the three equations of motion and its application to solve these types of equations.