Question

An engine of a vehicle can produce a maximum acceleration of $4{\text{ m}}{{\text{s}}^{ - 2}}$. Its brakes can produce a maximum retardation of $6 m{s^{ - 2}}$. The minimum time in which it can cover a distance of $3km$ is:
$A.{\text{ 30s}} \\\ B.{\text{ 40s}} \\\ C.{\text{ 50s}} \\\ D.{\text{ 60s}} \\\$

Answer 1

The change in the velocity which may increase or decrease with respect to time is known as the acceleration. The retardation is the act of slowness or the delay. It is the negative acceleration. Here, first convert all the units of the given terms in the same format and then solve placing values in the formula.

Complete step by step answer:
Let the maximum acceleration be $= \alpha$
The maximum retardation be $= \beta$
Distance be $= s$
Given that:
$\alpha = 4m/{s^2} \\\ \Rightarrow \beta = 6m/{s^2} \\\ \Rightarrow S = 3km = 3000m \\\$
$\Rightarrow S = \left[ {\dfrac{{\alpha \beta }}{{2(\alpha + \beta )}}} \right] \times {t^2}$ $$
Substitute the given values in the above equations
$\Rightarrow 3000 = \left[ {\dfrac{{4(6)}}{{2(4 + 6)}}} \right] \times {t^2}$
Simplify the above right hand side of the equation
$\Rightarrow 3000 = \left[ {\dfrac{{24}}{{2(10)}}} \right] \times {t^2} \\\ \Rightarrow 3000 = \left[ {\dfrac{{24}}{{20}}} \right] \times {t^2} \\\$
Make unknown “t” as the subject
$\Rightarrow {t^2} = \dfrac{{3000 \times 20}}{{24}}$
Simplify the above equation –
$\Rightarrow {t^2} = 2500$
Take square root on both the sides of the equation –
$\Rightarrow \sqrt {{t^2}} = \sqrt {2500}$
Square and square-root cancel each other on the left hand side of the equation.
$\Rightarrow t = \sqrt {2500} \\\ \therefore t = 50s \\\$
Thus, the required answer is - The minimum time in which it can cover a distance of $3km$ is $50s$.

Hence, from the given multiple choices – the option C is the correct answer.

Note: Always remember the units, derived units and SI units of the physical quantities to get the relation between the two or more physical quantities. Always check the given units and the units asked in the solutions. All the quantities should have the same system of units. There are three types of the system of units.
-MKS System (Metre Kilogram Second)
-CGS System (Centimetre Gram Second)
-System International (SI)
Also, remember the conversional relations among the system of units to make all the given units in the same format.