# How Newton's Laws Relate to the Law of Conservation of Energy & Momentum

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

The many different topics under physics can be connected in ways you might not expect. Here you'll learn Newton's second and third laws help us to derive conservation of energy and momentum.

## Connections Between Physics Topics

For teaching purposes, physics is split into different categories. There are the main topics like Classical Mechanics, Thermodynamics, Electricity and Magnetism, and Quantum Mechanics. Even these large topics are broken down further into smaller ones. For example, in Classical Mechanics you'll learn about linear motion, rotational motion, forces, energy, and momentum all as subtopics.

It's easy to think of these subtopics as completely separate concepts that have nothing to do with each other, but this isn't true. Physics teaches us how the natural world works around us, and all these concepts combine to form that understanding.

These topics are often connected in ways we might not even realize. Let's see how Newton's laws of motion relate to two examples, conservation of energy and conservation of momentum.

## Conservation of Energy

For conservation of energy we're going to focus on Newton's second law, which tells us that the acceleration of an object is directly proportional to the net force acting on it, and indirectly proportional to its mass.

This law can be used to help show that the law of conservation of energy is true. To do this, let's start with reviewing what exactly that law entails.

The law of conservation of energy states that the total energy in an isolated system remains constant over time. Mathematically we can write this out as the total energy at time two (E2) minus the total energy at time one (E1) divided by the change in time between the two (Δt) equals zero.

Now, remember that total energy equals kinetic energy (KE) plus potential energy (PE).

We've rearranged the equation so that the left-hand side consists of two separate parts, one for KE and one for PE. In order to show that the conservation of energy is true, we're going to show that the left-hand side does in fact equal zero. Let's start by looking at only the KE part of the left-hand side.

Recall that KE = (1/2)mv2 where m is mass and v is velocity.

With some algebra we can find that (v22 - v12) = (v2 - v1)(v2 + v1).

Here (v2 + v1)/2 is average velocity (v), and (v2-v1) / Δt is a change in velocity over time, which is acceleration (a).

This is where Newton's second law finally comes into play. It tells us that F = ma, so the kinetic energy portion of our equation can be written as:

Now, let's move onto the potential energy portion of the equation.

PE2 - PE1 is a change in potential energy (ΔPE). A change in potential energy is related to work (W) and force through two formulas:

Combining these two formulas with the potential energy portion of the equation results in:

A change in position (Δx) divided by a change in time (Δt) is equal to average velocity.

Finally, we take what we found and insert it back into our original formula.

We found that the left-hand side of the equation equals the right-hand side. With the help of Newton's 2nd law, we've shown that the conservation of energy holds true.

## Conservation of Momentum

So how does Newton's laws relate to conservation of momentum? Well, look at Newton's third law, which informs us that when one object enacts a force upon a second object, that second object also enacts a force back in equal magnitude but in the opposite direction.

One good example of this is a target shooter firing their pistol. The pistol exerts a force of the bullet pushing it forward, and the bullet also exerts a force back on the pistol that is felt by the shooter as recoil.

Getting back to conservation of momentum, let's start by calling the force object one enacts on object two F1, and the force object two enacts back on object one F2.

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