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.

Some examples of topics covered under physics.
physics topics

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.


Newton


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.

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.


Newton


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.

Recoil forces the pistol back and up in the air as it is fired.
pistol 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.


momentum part1


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