The slope of the kinetic energy-displacement | vedclass

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(C) The kinetic energy $E$ of a particle of mass $m$ moving with velocity $v$ is given by $E = \frac{1}{2}mv^2$.To find the slope of the kinetic energ

Vedclass · Accepted Answer

Directly proportional to the acceleration

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(C) The kinetic energy $E$ of a particle of mass $m$ moving with velocity $v$ is given by $E = \frac{1}{2}mv^2$.To find the slope of the kinetic energy-displacement curve,we differentiate $E$ with respect to displacement $x$:$\frac{dE}{dx} = \frac{d}{dx}(\frac{1}{2}mv^2) = \frac{1}{2}m 	imes 2v 	imes \frac{dv}{dx} = mv \frac{dv}{dx}$.Using the chain rule,we know that $\frac{dv}{dx} = \frac{dv}{dt} 	imes \frac{dt}{dx} = a 	imes \frac{1}{v}$,where $a$ is the acceleration.Substituting this into the expression for the slope:$\frac{dE}{dx} = mv 	imes (\frac{a}{v}) = ma$.Since $m$ is constant,the slope $\frac{dE}{dx}$ is directly proportional to the acceleration $a$ (specifically,it is equal to the force $F = ma$ acting on the particle).Thus,the correct option is $C$.

The slope of the kinetic energy-displacement curve of a particle in motion is

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