Question

A block of mass 10 kg is moving in x-direction with a constant speed of 10 m/s. It is subjected to a retarding force $F = 0.1\,x\,J/m$ during its travel form $x = 20\,m$ to $x = 30\,cm$ . Its final kinetic energy will be:
(a) 475 J
(b) 450 J
(c) 275 J
(d) 250J

Answer 1

Hint:Some of the work is done by the retarding force applied on block which is equal to change in kinetic energy according to work energy theorem. From this, final kinetic energy can be found.

Complete step by step answer:
According to work energy theorem amount of work done is equal to change in kinetic energy.
$Work\,done = \left[ {final\,\,kinetic\,\,energy} \right] - \left[ {initial\,\,kinetic\,\,energy} \right]$ … (i)
Let initial kinetic energy$ = \dfrac{1}{2}m{v^2}$ … (ii)
$\therefore \,\,m\left( {mass} \right) = 10\,kg$
$v\,\left( {initial\,\,speed} \right) = 10m/s$
Use given value in equation (ii)
${\left( {K.E.} \right)_{initial}} = \dfrac{1}{2}\left( {10} \right)\,{\left( {10} \right)^2}$
${\left( {K.E.} \right)_{initial}} = 500\,joule$ … (iii)
To calculate final kinetic energy need to calculate work done by using the calculate method.
Let a small amount of work be done displacing the particle by dx.
$dw = Fdx\,\,\cos \,Q$ … (iv)
$\therefore \,\,F = 0.1\,x\,J/m$
$Q = {180^o}$(force is retarding)
Hence$\cos {180^o} = - 1$
$dw = - 0.1\,x\,dx$. … (v)
Integrate equation (v) by using limits of displacement${x_1} = 20m$ to ${x_2} = 30m$
$\int {dw} = - \int\limits_{{x_1} = 20}^{{x_2} = 30} {0.1\,x.dx.} $
$w = - \,0.1\left| {\dfrac{{{x^2}}}{2}} \right|_{{x_1} = 20}^{{x_2} = 30}$
$ = - \,.05\left[ {{{\left( {30} \right)}^2} - {{\left( {20} \right)}^2}} \right]$
$ = - \,.05\left[ {900 - 400} \right]$
$ = - \,.05\left[ {500} \right]$
$W = - \,25\,Joule$ … (vi)
Put the value's of initial kinetic energy and work done in equation (i)
$ - 25 = {\left( {K.E.} \right)_{final}} - 500$
${(K.E.)_{final}} = - 25 + 500$
${(K.E.)_{final}} = 475\,\,joule.$

Note:Under retarding force Kinetic energy decreases this is due to fact that the retarding force results in decrease in velocity. That is why even the work done by retarding force comes out to be negative.