Question

The equation of motion of a car is $s=t^2–2t$, where is measured in hours and s in kilometres. When the distance travelled by a car is $15km$, the velocity of the car is

• A. $2km/hr$
• B. $4km/hr$
• C. $3km/hr$
• D. $8km/hr$

Answer 1

The correct option is D

$8km/hr$

Step 1: Given Data:

Distance$\left(s\right)=15km$

The equation for the motion of the car, $s=t^2-2t·····\left(1\right)$

Step 2: Formula used:

Velocity is defined as rate of change of displacement. that is-

$v=\frac{ds}{dt}$

Step 3: Calculation of the velocity of the car:

The distance travelled by car is $15km$. Substitute the value of distance in equation $\left(1\right)$ to obtain the time taken to travel this distance.

$$\begin{gathered} t^2-2t=15 \\ {t}^2-2t-15=0 \\ {t}^2-5t+3t-15=0 \\ t\left(t-5\right)+3\left(t-5\right)=0 \\ \left(t-5\right)\left(t+3\right)=0 \\ t=5,-3 \end{gathered}$$

$-3$ can be neglected because time can't be negative. So

$t=5hrs$

Now differentiate the equation$\left(1\right)$ with respect to time to get the velocity.

$\begin{array}{rcl}\frac{ds}{dt}& =& 2t-2\\ v& =& 2t-2.......\left(2\right)\end{array}$

Substitute the value of $t=5hrs$ in equation (2) to get the velocity-

$$\begin{gathered} v=2\times 5-2 \\ v=8\raisebox{1ex}{$km$}\!\left/ \!\raisebox{-1ex}{$hr$}\right. \end{gathered}$$

Hence, option D is the correct answer.