Derivation of Work-Kinetic Energy Theorem

       for special case of a constant net force acting on an object of mass m moving in a straight line..

 

If the net force is constant, the acceleration must be constant from Newton’s Second Law:

      Eq 1

If the acceleration is constant, from Kinematics we have

      Eq. 2

It is convenient to multiply Equation 2 by m/2:

      Eq. 3

Define the Kinetic Energy, K as K = (1/2) m v2 and use Newton’s Law:

      Eq. 4

Define the net work done on the object as

W net = Fnet(x — x0)       Eq. 5

Using the definitions in Equations 4 and 5 in Equation 3, we get

K = K0 + Wnet       Eq. 6

which is called the Work Kinetic Energy Theorem.

 
On a graph of net force versus position, with a constant net force, the work, Eq. 5, is just the area under the curve. This will be true even when the force is not constant.

 

Consequences of Work-Kinetic Energy Theorem

1. On a graph of net 1D force versus position, the area under the between two positions is the net work done on the object.

Remember that work is measured as area from axis to the line and can be positive or negative.

2. If the net work done on a point object is positive, the object is moving faster at the end than at the start.