Question

A particle of mass $5\text{kg}$ moving in the $X-Y$ plane has its potential energy given by $U=(-7x+24y)\text{J}$. The particle is initially at the origin and has a velocity, $u=(14.4\hat{i}+4.2\hat{j})\text{m/s}$. Then,

• A. The particle has speed $20\text{m/s}$ at $t=4\text{sec}$
• B. The particle has an acceleration $25{\text{m/s}}^2$
• C. The acceleration of the particle is normal to the initial velocity
• D. None of the above are correct

Answer 1

The correct option is C The acceleration of the particle is normal to the initial velocity

Force acting on the particle
$\overrightarrow{F}=-\frac{dU}{\partial x}\hat{i}-\frac{dU}{\partial y}\hat{j}=7\hat{i}-24\hat{j}$
Acceleration of particle
$\overrightarrow{a}=\frac{\overrightarrow{F}}{m}=\frac{7\hat{i}-24\hat{j}}{5}$
$\Rightarrow |\overrightarrow{a}|=\frac{\sqrt{7^2+{24}^2}}{5}=\frac{25}{5}=5{\text{ms}}^{-2}$

Velocity of particle is given by
$\overrightarrow{v}=\overrightarrow{u}+\overrightarrow{a}t$
$=(14.4\hat{i}+4.2\hat{j})+(1.4\hat{i}-4.8\hat{j})t$
Velocity at $4\text{sec}=20\hat{i}-15\hat{j}$
$\Rightarrow$ Speed at $4\text{sec}$ $=\sqrt{(20{)}^2+(-15{)}^2}=25{\text{ms}}^{-1}$
Only option (c) is correct.

Now $\overrightarrow{a}.\overrightarrow{u}=(1.4\hat{i}-4.8\hat{j}).(14.4\hat{i}+4.2\hat{j})$
$=20.16-20.16=0$
$\therefore$ The acceleration of particle is normal to its initial velocity.