Just a few days ago, in the Trajectories post, I could not get a simple example to work. To recap: the expected result was 0.2 metres and I was getting exactly 0.0000
The following day I quickly worked out what it was. I was merging two ‘worked examples’ and had missed one little change. Normally that wouldn’t matter, but coincidences in their values chosen – no doubt to make the maths easier – meant I kept getting 0. By hand or in Excel.
First Example I Used
The first example, doing the maths one way, was a projectile fired at 10 m/s and 45 degrees. They set the value of Earth’s gravity (G or g) at 10, which is slightly higher than the real value, but hey.
By then I’d found the second example which gave me the ‘after 10 metres along it should be 0.2 metres off the ground’ values to check.
So I did the first way by hand, hoping to get 0.2:
Ok, not good. So I went back and looked at…
The Second Example
A program, doing it a different way (aka calculus). It was only then that the light went on. They had set G to be the more exact value of 9.8, not 10. Yeah, it’s not obvious, but jumped out at me:
The program correctly calculated the 0.2 answer.
But So What?
Big deal; 10 or 9.8. That’s like 2% difference.
But look back at the hand calculations. A lot of the values were 10: G (g), Vo and x. I then worked out that they effectively cancelled each other out in the maths.
By using those, and also picking 45 degrees** (see below) that last calculation turned into: 10-10 which is 0.000000 🙂
By changing G(g) to the correct value of 9.8 (and getting the ‘sign’ of g right), things didn’t cancel each other out and the correct answer popped out:
The Irony of Advanced Maths
One of the ironies of doing advanced maths and physics at uni was you learnt more skills on how to do quick estimates. This was useful as you could first estimate the ‘sort of’ value you hoped to get when you did the ‘real’ maths. I still use it today as means of cross-check aka sanity check.
It was, of course, part of the reason the above came unstuck. G is 9.8 not 10. But, as in the Virus post, having things in units of 10 makes the maths easier. In this example, the three things being 10 made it easier in one example, but blew up another.
** A further thing that nuked me was the 45 degrees. In the ideal case (no air resistance, level surface etc) throwing it a 45 degrees will hurl the projectile the maximum distance. Probably why they picked it.
But here, cos2(45) is 0.5 and when we then multiple that by 2, it turns into 1 and it too effectively vanishes in terms of impact, when used in further multiplication! 100 x 1 is still 100.
If it had been 30 degrees: cos2(30) multiplied by 2 would be 1.5 and so wouldn’t ‘vanish’