An object of mass equal to 1 kg is under the action of a force resulting given by \(\displaystyle F=-3x-5x^2\) where F is given in newtons. (a) What is the potential energy of an object at x = 2 m? (b) If the object has a speed equal to 4 m / s in the negative sense when it is in position x = 5 m what is its speed as it passes through the origin?

Answer:

(a) 19,3 J

(b) 22,5 m/s

a) Hint: \(\displaystyle \Delta V = \int F~dx\)

You technically need to define a zero point level for this potential energy function, so all you can find is the change of potential energy between two points. However you can easily see, if you integrate the function as an indefinite integral (and ignore the arbitrary constant) there is already a "built in" zero point. This is how you are going to get your solution.

b) Another hint: The force here is not required to be conservative so you can't apply conservation of energy directly. But you can still say that

\(\displaystyle W_{nc} = \Delta E = \Delta K\)

(presumably there is no vertical displacement) and use the given force as a non-conservative force. So

\(\displaystyle \int_5^0 (-3x - 5x^2)~dx = \Delta K\)

I'll leave it to you to determine why there isn't a negative sign in front of the integral.

-Dan