
Substitution is given a physical meaning.

In physics, we take measurable quantities from the real world, and attempt to find meaningful relationships between them. A basic example of this would be the physical ideal of 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
To get a feel for what a newton is, consider this: if an apple has a mass of $0.1\unit {kg}$, what force would an apple exert on your hand due to the acceleration due to gravity?

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 $0.1\unit {kg}$ and it is dropped from a height of $1\unit {m}$, then approximately $1$ joule of energy is released when it hits the ground. Let’s see if we can explain why this is true.

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.

Which of the following are examples where work of this kind is being done?
studying calculus a car applying breaks to come to a stop over a distance of $100\unit {ft}$ a young mathematician climbing a mountain a young mathematician standing still, holding a $1000$ page calculus book for $10$ minutes a young mathematician walking around with a $1000$ page calculus book a young mathematician picking up a $1000$ page calculus book
We can write the definition of work in the language of calculus as, The SI unit of work is also a joule. To help understand this, $1$ 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.

The Work-Energy theorem says that:

This could be interpreted as:

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.