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

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) x2 (B) ex (C) x (D) loge x

Tardigrade Admin · Accepted Answer

From given information a = - Ax, where a is acceleration, x is displacement and k is a proportionality constant.

\frac{v d x}{d x} = - k x

\Rightarrow v d v = - k x d x

Let for any displacement from 0 to x, the velocity changes from

v_{0} t o v

\Rightarrow \int_{v_{0}}^{2} v d v = - \int_{0}^{x} k x d x

\Rightarrow \frac{v ^{2} - v _{0}^{2}}{2} = - \frac{k x ^{2}}{2}

\Rightarrow m (\frac{v ^{2} - v _{0}^{2}}{2}) = - \frac{mk x ^{2}}{2}

\Rightarrow Δ K \propto x^{2}

(Δ K i s l oss in K E)

Q. 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

$x^{2}$

$e^{x}$

x

$l o g_{e} x$

Q. 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

x2

ex

x

loge​x

Q. 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

$x^{2}$

$e^{x}$

$l o g_{e} x$