Question

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to

• A. $x$
• B. $e^x$
• C. $x^2$
• D. $log_e\,x$

Answer 1

Answer: (C)

$x^2$

Answer 2

Find the loss of kinetic energy for any displacement x is proportional to.
Let,
Net work done by all the forces gives the change in kinetic energy
⇒ $W=\frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$
​ W=$k_f-k_i$
​where,
m= mass of the body
v0= initial velocity
v= final velocity
Retardation aα−x
⇒ $\frac{dv}{dt}$ =−kx [k is a proportionality constant]
⇒ $v\frac{dv}{dx}$ ​=−kx
⇒ $\int_{v_1}^{v_2}vdv=-k\int_{0}^{x}xdx$
⇒ $\frac{1}{2}v_2^2-v_1^2 =-\frac{1}{2}kx^2$
Loss in kinetic energy =$\frac{1}{2}m(v_2^2-v_1^2)=-\frac{1}{2}kx^2$
Loss in kinetic energy $\alpha x^2$
Therefore, The correct option is 'C' is $x^2$.