Thursday, 4 November 2021

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.
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.


James Higham said...

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

Dem lyrics iz pure poetry.

Mark Wadsworth said...

JH, thanks, occasionally I have a lucid moment.

Lola said...

I am so pleased that someone else has the 'boring meeting' / 'insomnia' issue.

My solutions to the latter are to think through the next stages in the car I am building and how to make the bits and or fit the bits, or to go over economic arguments. As to the former, as I am A Boss I usually just exit, or revert to Plan A - the same as for insomnia.

On the maths kick, a few nights I was trying to work out the right spring rates for the car....

Mark Wadsworth said...

L, do you ever have blinding insights while doing so, like "I could take the front springs from car A and use them as rear springs for car B"?

Bayard said...

I have always found that the extra accuracy you get when you take g to be 9.81 is not worth the extra hassle in calculation over taking g as 10.

Lola said...

MW Yes! Fr'instance the existing rear spring damper units are off a Hillman Imp...

But I am now getting clever and looking at the whole system - try to visualise all the components their locations and how they move. e.g. thew wheel centre line is outboard of the spring centre line by about 6 inches. This creates a lever which makes the spring effectively softer than its actual rate, plus the spring is canted by about 10 degrees.

Fun eh?

Lola said...

MW - I meant to say the rear spring damper units on the rear of the car I am re-building are the front units off a Hillman Imp

Mark Wadsworth said...

B, I prefer 9.8, give gravity a chance. If you do ten, you might end up with the right answer by mistake because you dropped or added a zero somewhere and also forgot to divide or multiply by g.

L, ha! That was a lucky guess on my part.

Lola said...

MW. It's how 'specials' have always been made.

Dr Evil said...

32 feet/sec/sec

Mark Wadsworth said...

Dr E, fair point. But I use SI units, so much easier and internationally accepted.

Scrobs. said...

I've been looking at two equations just yesterday!

1) What is the percentage increase in cost for an expensive beer from Waitrose, from an expensive beer from Tesco.

I tried to do this while driving the car, but had to do a quick spreadsheet when I got home!


My VW Golf does approximately 43 mpg, but with a bit of 'featherfooting' (it's automatic), I can get it up to 44.2 mpg.

How much am I saving on premium petrol, (I do mainly short journeys, and don't use Wynns)! It's currently 16p a mile.

Again, I had to do another spreadsheet after failing to solve it in my head!

I'm applying for 'Nerd of the year' it seems...

Mark Wadsworth said...

Sc, if feather footing improves mpg by 3%, then it saves you half a penny a mile (3% of 16). There's no need to be any more scientific than that, because 43, 44.2 and 16 are just mid-point estimates.

Graeme said...

Are your kids doing sats, so they need to know these shortcuts?

Mark Wadsworth said...

G, my kids are already at uni, so it's a bit late for that.

Scrobs. said...

Thanks Mark!

In fact, it went up to 44.7 mpg yesterday, so as the A21 is renowned for that average speed, I'll rest easier!