Equating Energies and the VATS

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There are various forms of energy, however two of the most principal mechanical energies are kinetic energy and gravitational potential energy:

  • Kinetic energy refers to the energy used in an object's motion;
  • Gravitational potential energy refers to the potential energy of an object when at a given height.

Assume you have a steel ball at rest on the top of a cliff - it will eventually be released and fall to the ground in linear motion, not bouncing upon impact. A possible question would be:

Calculate the height/mass/velocity of the ball.

To do this, you can assume the gravitational potential energy of the ball is fully transferred into the ball's kinetic energy store. At first glance, this makes sense - after all, the ball can only ever move - it's not being stretched, electricity isn't being conducted through it, the like. There is the issue of resistive forces such as air resistance and we can take it into account. For the time being, I'll ignore these forces as they make the problem more complicated than it has to be. 

The formula for kinetic energy is Ek = (1/2)mv2, whereas the formula for gravitational potential energy is E = mgh. The most straightforward thing to do is to equate the two formulae, and if you do so, some simple cancelling leads you to this very pleasant formula:

v2 = 2gh

I remember discussing the answers we put in a physics exam, and my friend took this approach. I meanwhile opted to use the SUVAT formulae which are commonly used for objects undergoing constant acceleration - more on them later. A few days ago, however, I realised that this formula is simply a SUVAT in disguise - it's v2 = u2 + 2as, except the object started at rest (so u = 0), and the object falls at a constant acceleration of g, or about 9.81 ms-2. 

From here, I opted to mess around with the SUVAT formulae. There are five of these, all linked through their use of displacement (s), initial (u) and final (v) velocities, constant acceleration (a) and time (t). They can only ever be used when the acceleration is constant - in other words, they can only be used when the force acting on an object doesn't vary, as otherwise the acceleration will vary (in accordance with Newton's Second Law, which often includes a ∆ for this reason in front of the F and a). Removing the initial velocity from all expressions and making the acceleration g, I found what I will affectionately call the VATS. They're nothing groundbreaking, indeed they've been known in physics for a while now, but I like the term regardless.

  1. Take v = u + at and use the rules, and you get v = gt.
  2. As before, v2 = 2gh
  3. Substitute the first equation into the second equation, and you end up with (gt)2 = 2gh. Cancelling out a g on both sides, you get gt2 = 2h, and simplify to get h = (1/2)gt2. This is obviously an altered s = ut + (1/2)at2.

I believe that's it, because when you alter the other SUVATs, you end up with equation 3 again. Therefore, those are three altered equations that can be used, so long as an object is falling at free fall and started at rest - conditions which make them less appealing to use.

In reality, steel balls may not always fall linearly - instead, they may be projected from a certain point, at which point the initial velocity is no longer 0. However, I did enjoy finding the links between expressions and formulae which I've used so often, yet never truly considered how they all apply in the end.

I didn't discover anything truly revolutionary, all I did was alter some equations. Yet it did make me appreciate maths and physics even more, which I suppose is the most important bit.

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