This is a problem of variable force acting on the body

Now at x = 2m/s, v= 3m/s

Kinetic energy at intial point = 0

Thus, change in kinetic energy at x=2/s is = (1/2)mv²

or Change in KE = 0.5*2*9 = 9 J

Here work done on the body is converted into its kinetic energy

Thus work done = F.d

But force here is a function of displacement x as follows

dF/dx = (9-0)/(2-0) from theabove condition

 

or dF/dx = 4.5 N/m

Thus force at any position x is given by

Thus F(x) = 4.5 x Newton

Thus work done by the force on the body is given by

W = _{[ 0]}^{[ 10]} F(x).dx = _{[ 0]}^{[ 10]} 4.5 x dx = 4.5 * 10²/2 = 225 J

 

So kinetic energy gained by the object = 225J

 

or 1/2 m v² = 225

 

or v² = 225 or v = 15m/s

 

Thus final velocity of the at x = 10m is 15m/s