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.
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.
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.
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So, how do 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.
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.
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, according to the law of conservation of momentum, 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.
Starting from Newton's 3rd law, we have found the law of conservation of momentum.
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 demonstrate the conservation of energy, or that the total energy of an isolated system remains constant over time.
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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