# Deriving the Laws of Motion

I remember back in high school physics, a lot of my fellow students struggled with memorizing all the equations.

I was never very good at memorizing equations. I found it easier to remember how equations were derived instead.

These are the variables in the equations:

• `t` time an object has been moving.
• `v` velocity an object obtains after some time `t`
• `v₀` the initial velocity of the object at time `t = 0`.
• `r` distance traveled by object after time `t`.
• `r₀` initial distance traveled before we start counting time.
• `a` acceleration of object.

The first equation is probably the only one which is immediately obvious, so I will concentrate on showing how we arrive at the last two.

Here are the steps for how we arrive at equation number two:

1. We can use this equation of acceleration `a = 0`, meaning the velocity `v` is constant.
2. However the velocity `v` is a function of time, `v(t)`. Thus we got to integrate the velocity to arrive at the total distance traveled.
3. We insert the definition of velocity as a function of time.
4. Perform the integration.
5. Subtract the ranges from 0 to t. This may seem a bit confusing because I happen to use `t` both in the expression and as the last value in the range.
6. We arrive at the final equation for distance traveled as a function of time.

We can use the first and second equation for motion to derive the third one.

This steps are just regular algebra so there should be no need to explain them in detail.

My motivation for showing this is to work towards later stories about equations used in rocketry and how to the equations for work and kinetic energy are related.

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