According to work-energy theorem,
work done =change in kinetic energy of the vehicle
`:. W=K_(f)-K_(i) orF.dx=K_(f)-(1)/(2)mv_(i)^(2)`
or `F.dx=k_(f)-(1)/(2)xx10xx(10)^(2)or F.dx =k_(f)-500`
or `int_(x=20)^(x=30)(-0.1)x dx=k_(f)-500`
or `-0.1[(x^(2))/(2)]_(x=20)^(x=30)=K_(f)-500`
or `-0.1[((30)^(2))/(2)-((20)^(2))/(2)]=K_(f)-500`
or `K_(f)-500=-0.1(450-200)`
or `K_(f)-500=-25`
`:. K_(f)=500-25=475 J`

Correct Answer – D
Here, m=10 kg, `v_i= 10 m s^(-1)`
Initial kinetic energy of the block is
`K_i=1/2mv_i^2=1/2xx(10 kg)xx(10 m s^(-1))^2 =500 J`
Work done by retarding force
`W=intF_rdx=underset(20)overset(30)int -0.1 xdx = -0.1 [x^2/2]_20^30`
`= -0.1[(900-400)/2]= -25 J`
According to work energy theorem
`W=K_f-K_i`
`K_f=W+K_i = – 25 J + 500 J = 475 J `

Correct Answer – D
Here, `m=10 kg, u=10m//s, F=-0.1 x J//m`
`[` as force is retarding `]`
`x_(1)=20 cm, x_(2)=30 cm`
As `W=int_(x_(1))^(x_(2))Fdx=int_(20)^(30)=-0.1x dx=-0.1[(x^(2))/(2)]_(20)^(30)`
`=(0.1)/(2)[(30)^(2)-(20)^(2)]=-(0.1)/(2)[900-400]`
`=-(0.1)/(2)[500]=-25J`
`K.E_(i)=(1)/(2)m u^(2)=(1)/(2)xx10xx(10)^(2)=500J`
According to work `-` energy theorem,
`W=K.E_(f)-K.E_(i)` or `-25=K.E_(f)-500`
`K.E_(f)=475J`

When a force acts upon a moving body , then the kinekic energy of the body increase and the increase is equal to the work done. This is work energy therom.
Work done `= (1)/(2) mv^(2) – (1)/(2) mu^(2) = K_(f) – K_(i)`
Another definition of work is force xx displacement.
`:. Fdx = K_(f) – (1)/(2) xx mv_(1)^(2)`
where the subscripts `f` and l` stand for final and initial.
`F. dx = K_(f) – (1)/(2) xx10 xx (10)^(2)`
`implies F. dx = K_(f) – 500`
`rArr underset(x=20)overset(x=30)int=K_(f)-500`
Using the formula `int x^(n) dx = (x^(n+1))/(n+1)`, we have
`- 0.1 [(x^(2))/(2)]_(x=20)^(x=30) = K_(f) – 500`
`- 0.1 [(30)^(2)/(2) – ((20)^(2))/(2)] = K_(f) – 500`
`implies K_(f) – 500 = – 25`
`implies K_(f) – 500 – 25 = 475J`

Correct Answer – A
`K_(f)-K_(i) =W =int Fdx`
`:. K_(f) =K_(i) + int_(20)^(30)(-0.1x)dx`
`1/2 xx 10 xx (10)^(2)-[0.1(x^(2))/(2)]_(20)^(30)`
`475 J`.