I am confused and don\'t know how to get the following answers for each question

Question

I am confused and don't know how to get the following answers for each question.

Could someone please explain how to get the following answers?

I will rate 5 star!!

Consider a classical ball that is initially stationary. and located at x = 0, on a potential energy surface of the form V(x) = x^2 - x + 1 (in J units). You may assume that the ball does not experience any friction.

a). What is the initial force on the ball along the x-direction (in SI units)?

     Ans. 1 J/m

b). What is the total energy of the ball (in SI units)?

     Ans. 1 J

c). What is the maximum value of the kinetic energy that the ball will attain after it is released.

     Ans. 1/4 J

Explanation / Answer

a) initial x=0 , so V(x) = V(0)=1 J ,

so initial force = 1J/m

b) since initailly kinetci energy =0 , Total energy = V(x) = 1 J

c) max KE will be possible when potentilal energy is minimum ,

V'(x) = 2x-1, V''(x) = 2 , increasing function , so whgen V'(x) = 0 = 2x-1, x = 1/2 = 0.5 ,

V(0.5) = 0.5^2 -0.5+1 = 0.75 J

now KE = TE -V = 1-0.75 = 0.25 J = 1/4 J

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