Potential energy of a body of mass 1 kg free to move along X - axis is given by ` U(x) = (x^(2)/2 - x) J`. If the total mechanical energy of the body is 2 J , then the maximum speed of the body is (Assume only conservative force acts on the body )

Options

A. `sqrt(5) ms^(-1)`
B. `5 ms^(-1)`
C. `3.5 ms^(-1)`
D. `sqrt(8) ms^(-1)`

Answer

A

Explanation

Key Idea Total mechanical energy of a system is the addition of potential and kinetic energy
`E = U + KE`
Here, mass of the body, `m = 1 ` kg
`E_("mech") = 2 J `
So, for `U_(min) = (dU(x))/dx= d/dx [x^(2)/2 - x] =0`
`rArr x - = 0`
` rArr x = 1 `
Hence, `U_(min) = (1)^(2)/2 - 1 = -1/2 `
Kinetic energy, `KE = 1/2 mv^(2) = v^(2)/2 (because m = 1 kg)`
Now, putting the values in above expression
`rArr 2 = -1/2 + v^(2)/2`
` rArr v^(2) = 5`
` rArr v = sqrt(5) m//s`
Hence, the correct option is (a)