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

Lesson Transcript
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 how Newton's second and third laws help us to derive conservation of energy and momentum. Updated: 07/08/2019

Connecting 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 of 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.

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  • 0:04 Connecting Physics Topics
  • 0:55 Conservation of Energy
  • 3:35 Conservation of Momentum
  • 5:12 Lesson Summary
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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.

energy part1

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

energy part2

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.

energy part3

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

energy part4

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

energy part5

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

energy part6

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:

energy part7

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

energy part8

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

energy part9

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

energy part10

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

energy part11

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

energy final

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.

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