Calculating the speed of a falling object etc. using the physics approach.

I like doing a bit of mental arithmetic, when I'm in a boring meeting or lying awake at night. A typical challenge is calculating the speed of falling objects; how long it takes them to fall etc. There's one constant, acceleration due to gravity (hereafter abbreviated to 'gravity' for brevity) = 9.8 m/s2 and then you have to work out how to work it out; then remember how to work it out while you actually work it out.

It's surprisingly fiddly, tedious and not much fun. Here's a link to an explanation with an embedded calculator.

It occurred to me this morning that taking the maths approach is a load of bollocks, it's quicker, easier and simply more fun taking the physics approach. You just have to remember a bit of GSCE level physics:
1. Initial potential energy of an object = kinetic energy of the object just as it hits the ground.
2. Potential energy = mass x height x gravity.
3. Kinetic energy = half x mass x velocity squared.

For simplicity, mass is always 1kg so does not appear in the answers (it would cancel out anyway), we're using SI units and we're ignoring air resistance.

Q1: Object is doing 60 m/s when it hits the ground. From what height was it dropped?

A: Closing KE = 1/2 x 60 x 60 = 1,800
∴ Starting PE = 1,800
∴ Starting height = 1,800/9.8 = 183 metres

Q2: An object is dropped from a height of 500 metres,
a) at what speed does it hit the ground
b) how long before it hits the ground?

A: Starting PE = 9.8 x 500 = 4,900
∴ Closing KE = 1/2 x 9,800
∴ Closing velocity squared = 9,800
∴ a) Closing velocity = 99 metres/second
(Calculating square roots made easy here)
∴ b) Time taken to fall (constant acceleration at 9.8 m/s2) = 99/9.8 = 10.1 seconds.
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The old fashioned maths approach is probably better (simpler calculation and more intuitive) if you are told time taken to fall:

Q3: An object falls for five seconds before it hits the ground. From what height was it dropped?

A: Closing velocity = 5 x 9.8 = 49
∴ Average velocity = 1/2 x 49 = 24.5
∴ Height = 5 seconds x 24.5 = 122.5 metres

The physics approach would be:
A: Closing kinetic energy = 1/2 x (5 x *9.8) x (5 x 9.8) (no need to calculate the actual number)
Starting potential energy = closing kinetic energy
Height = starting potential energy ÷ *9.8
∴ Height = 1/2 x 5 x 5 x 9.8 = 122.5 (the *9.8s cancel out)

Both approaches boil down to:

Height = 1/2 x time in seconds squared x gravity, but you'd have to remember this extra equation, so this approach is not advised.