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

The Correct Option is C

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 \(x^2\).