Question

Derive law of conservation of momentum from newtons second law

Answer 1

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