Question

A car moving at a speed $v$ is stopped by a retarding force F in a distance s. If the force of the car is 3F, the car stops in a distance.
A) $\dfrac{s}{3}$.
B) $\dfrac{s}{6}$.
C) $\dfrac{s}{9}$.
D) $\dfrac{s}{{12}}$

Answer 1

The formula of force can be used for solving this problem also relations in Newton's law of motion can be used to solve this problem. The force on anybody depends on the mass of the body as well as the acceleration of the body.

Formula used: The formula used for force is given by $F = m \cdot a$ where F is force m is mass and a is acceleration.

Complete step by step answer:
As it is given that retarding force is given by $F = m \cdot a$ where F is force m is mass and a is acceleration,
As the retarding force is F therefore the acceleration is given by,
$\- F = m \cdot a \\\ a = - \dfrac{F}{m} \\\$
Applying the Newton’s law of motion relation,
${v^2} - {u^2} = 2as$………eq (1)
Where v is final velocity, u is initial velocity, a is acceleration and s is displacement.
The acceleration is given by $a = - \dfrac{F}{m}$ replace the value in equation (1),
Since the body is stopped by force F, therefore the final velocity becomes zero.

{v^2} - {u^2} = 2as \\\ \- {u^2} = 2s\left( { - \dfrac{F}{m}} \right) \\\ {u^2} = \dfrac{{2Fs}}{m} \\\ $$………eq (2) Now if the retarding force applied is 3F then, $ \- 3F = m \cdot a \\\ a = - \dfrac{{3F}}{m} \\\ $ Replace the value of the acceleration in equation (1) also the final velocity becomes zero as the applied force is retarding.

{v^2} - {u^2} = 2a{s_1} \\
- {u^2} = 2{s_1}\left( { - \dfrac{{3F}}{m}} \right) \\
{u^2} = \dfrac{{6F{s_1}}}{m} \\

Replace the value of ${u^2}$from equation (2) to equation (3),

{u^2} = \dfrac{{6F{s_1}}}{m} \\
\dfrac{{2Fs}}{m} = \dfrac{{6F{s_1}}}{m} \\
{s_1} = \dfrac{s}{3} \\

$$**So the correct answer for this problem is option A.** **Note:** Students should remember that the formula used for solving these types of problems also students should understand and remember all the relations included in Newton’s law of motion. While solving the relation of force we have taken negative sign because the force applied is retarding force which will result in decreasing of acceleration of the body and therefore the negative sign is taken.$$