Substitution is given a physical meaning.

**force**. Force applied to an object changes the motion of an object. Here’s the deal though, at a basic level and while we can put a physical interpretation to this arithmetical definition, at the end of the day force is simply “mass times acceleration.” The SI unit of force is a

**newton**, which is defined to be

In a similar way, the idea of **kinetic energy**, is “energy” objects have from motion.
It is defined by the formula The SI unit of energy is a **joule**, which is defined to be
To get a feel for the “size” of a joule, consider this: if an apple has a mass of
and it is dropped from a height of , then approximately joule of energy is
released when it hits the ground. Let’s see if we can explain why this is
true.

Now we need to know how long it takes for the apple to hit the ground, after being dropped from a height of meter. For this we’ll need a formula for position. Here , so we’ll need to use an indefinite integral:

Since , write with me hence and . Solving the equation for tells us the time the apple hits the ground. Write with me

So the apple hits the ground after seconds. Finally, the formula for kinetic energy is

Ah! So the kinetic energy released by an apple dropped from a height of meter is approximately joule.

Finally **work** is defined to be accumulated force over a distance. Note, there must be
some force *in the direction* (or opposite direction) that the object is moving for it to
be considered *work*.

On the other hand, a car applying brakes is a change in motion, and hence a force is applied. Since this force is applied over a distance, work is done.

Climbing a mountain is also an example of work, as one is applying force to overcome the acceleration due to gravity, over the distance that one is climbing.

No work is done when holding a calculus book, as there is no accumulated force over a distance.

It is also the case that no work is done when one walks around with a calculus book, this is because the “force” is in a direction perpendicular to the motion.

Finally, when one picks up a calculus book, you are moving the book against the force due to the acceleration due to gravity. Hence work is done.

**joule**. To help understand this, joule is approximately how much work is done when you raise an apple one meter.

Let’s again see why this is true.

Now we have a question:

**Why do work and kinetic energy have the same units?**

One way to answer this is via the *Work-Energy Theorem*.

- represent position with respect to time,
- represent velocity with respect to time,
- represent acceleration with respect to position,
- represent the starting time,
- represent the ending time,

then we also have that

- represents the starting position, ,
- represents the ending position, ,
- represents the starting velocity, ,
- represents the ending velocity, .

Now write with me, here we are working with functions of distance. We will use the substitution formula,

The Work-Energy theorem says that:

**The accumulated force over distance is the change
in kinetic energy.**

Moreover, this answers our initial question of why work and kinetic energy have the same units. In essence, energy powers work.