Question

An engine supplies a constant power ‘p’ to an automobile of mass m starting from rest. At an instant of time t

• A. $v\propto t^{\frac{1}{2}}$
• B. $v\propto t^{\frac{3}{2}}$
• C. $s\propto t^{\frac{3}{2}}$
• D. both A and B

Answer 1

Answer: (B), (C)

$s\propto t^{\frac{3}{2}}$

Step (1): Finding the relation between $v$ and $t$

When an automobile of mass $m$ accelerates from rest when the engine supplies a constant power $p$, then its velocity at a time $t$ is given by $v=\sqrt{\frac{2pt}{m}}$

From the above expression, we can observe that

$$\begin{gathered} v\propto \sqrt{t} \\ \Rightarrow v\propto t^{\frac{1}{2}} \end{gathered}$$

Step (2): Finding the relation between the displacement $s$ and time $t$

Let us take $v=K\times t^{\frac{1}{2}}(whereKisaconstant).$

We know that velocity $v$ is the derivative of displacement $s$ with respect to time $t$

Hence, $\frac{ds}{dt}=K\times t^{\frac{1}{2}}$

By applying integration on both sides, we get

$$\begin{gathered} s=J\times t^{\frac{3}{2}}(whereJisacons\tan t) \\ \Rightarrow s\propto t^{\frac{3}{2}} \end{gathered}$$

Hence, options B and C are correct.