The at Newton’s third law, which informs

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.

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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 (;t) equals zero.

energy part1


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.

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 (v22v12) = (v2v1)(v2 + v1).

Here (v2 + v1)/2 is average velocity (v), and (v2v1) / ;t 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:

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

energy part8


PE2PE1 is a change in potential energy (;PE).

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 (;x) divided by a change in time (;t) is equal to average velocity.

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.
1, and the force object two enacts back on object one F2.

momentum part1


Since these two objects are delivering their respective forces on each other simultaneously, the time it takes for each of the forces to act are equal.

To see how all this relates to conservation of momentum, we need to look at something called impulse (J). Impulse is the effect of a force acting on an object over some time period.

momentum part3


Earlier we found that the times between our two interacting objects were equal, and the forces they imparted were equal but in opposite directions. This means their impulses must also be equal and in opposite directions.

However, impulse is also equal to a change in momentum. So we can write the equation as change in momentum for object two equals and is opposite to the change in momentum in object one. This statement is also known as the law of conservation of momentum.


momentum final


Starting from Newton’s 3rd law we have found the law of conservation of momentum.

Lesson Summary

Various topics under physics may seem separate from each other, but there are connections between them that you might not realize. Two examples are the connections between Newton’s laws and conservation of energy and momentum.Newton’s second law states that the acceleration of an object is proportional to the force acting on it, and inversely proportional to its mass. This law can be used to help show that conservation of energy, which tells us that the total energy of an isolated system remains constant over time, is true.By starting with Newton’s third law, for every force there is an equal and opposite force, we can derive conservation of momentum, which tells us that a change in momentum in one of two objects colliding is equal and opposite to the change in momentum in the other colliding object.

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