The figure shows the action of a variable force 𝐹 on a body of mass 𝑚 kilograms as it moves the distance 𝑑. Given that 𝐹 is measured in newtons and 𝑑 is measured in meters, find the change in the body’s kinetic energy between 𝑑 equals zero meters and 𝑑 equals six meters.
We can call the change in the body’s kinetic energy ΔKE. And to solve for it using this diagram, it will be helpful for us to recall the work-energy principle. This principle tells us that the work that’s done on an object is equal to that object’s change in kinetic energy. So to solve for ΔKE, we only need to solve for the work 𝑊. We can recall that work is equal to force multiplied by distance.
And looking at our diagram, we see we have force on the vertical axis and distance on the horizontal. Since the work done is equal to the product of these two terms, graphically, that work is represented by the two shaded areas shown. The work is the area under the curve. Notice that at the point where 𝑑 is equal to one, the force crosses over from a positive to a negative value. This tells us that the work done in the first triangle, from 𝑑 equals zero to one, is positive work. While the work done on the object, from 𝑑 equals one to 𝑑 equals six, is negative work.
Let’s calculate these two terms and add them together to solve for the total work done, and therefore the total change in kinetic energy. If we start by considering the positive work done by the force 𝐹, that work, which we can call 𝑊 sub plus, is equal to one-half the base of the triangle, one meter, multiplied by its height, two newtons. Multiplying these terms together gives one joule. That’s the positive work done by the force. When we consider the negative work done, what we can call 𝑊 sub minus, we can calculate that work done by dividing the area under the curve beneath the 𝑑-axis into two shapes, a right triangle and a rectangle.
The area of the triangle, one-half its base of three meters times its height of negative six newtons, is added to the area of the rectangle. Its base is two meters and its height is negative six newtons. Altogether, the negative work done by the force 𝐹 is negative 21 joules. The total work done on the body is the sum of the positive and the negative work, or one joule minus 21 joules, or simply negative 20 joules. That’s the work done on, as well as the change in kinetic energy of, this body.