Using the slope of a graph to solve problems.

There are several quantities in mechanics which are the derivative of another quantity. Some are:

1) the net force on an object is the derivative of the momentum of that object with respect to (wrt) time,
2) the net force on an object is the derivative of that object's potential energy with respect to position.
3) velocity is the derivative of position wrt time
4) acceleration is the derivative of velocity wrt time

To see how to use this information, let's use #1 as an example. If you are shown a graph of momentum (y axis) v.s. time (x axis), then the force on that object is the slope of the line. For #2, it would be a graph of potential energy (y) v.s. position (x). An example of the latter is HRW problem #37 on page 192.

HRWP37p192

For part a, you need to know the initial kinetic energy so that you can compare it to the potential energy graph. If there is left over energy at 1 m (delta U - KE > 0) then that energy is kinetic and you can derive the speed for x < 2 m.
For part b, the negative of the slope of the line is the force. The sign gives you the direction (magnitude is always positive).

Remember that U + KE = constant (since no non-conservative forces are present). KE = .5 mv².