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Force and energy: a lack of understanding : coconuts

January 4, 2014

Alas, a lot of what I learnt at university and at school is lost. (One of Jane's occasional sayings is that one of her tutors said that education is what you retain after you have forgotten everything that you have learnt). I got an A at A level physics (note to my children: this was when A grades meant something) and then a top first in engineering science at Cambridge.

I think all this meant was at the time I had a great capacity for soaking up information, regurgitating it, applying it but without really understanding it. You get told a definition of two or three, shown some equations or principles, and then learn how to apply them.

Take, for instance, one of Newton's laws: F=ma. [note re predictive text: my iPad knows I am an accountant, and unhelpfully wants to change ma to M&A]. I can still apply it, and equations such as v=u+at, so am useful to daughter #2 with help with her physics homework. I have got a good understanding, at least on Earth, of acceleration, but do I really understand Force and mass? On the latter, can I clearly distinguish it from weight, and am I really clear about force and energy? Being sensible, of course I have some understanding, and can apply the separate concepts: but being honest, not fully.

The case of the falling coconut

Which would you rather be hit on the head by? A coconut falling from a tree height 3m, or from one with height 6m? Clearly, the former: but how strong is your preference? Does the 6m coconut hurt twice as much, or some other difference?

 

Each day, during our recent holiday in Maafushivaru, Maldives, we had to walk past or under several such trees. And one such bend, coconut corner, there was a 6m tree which shed four coconuts during our stay: once when we were walking past, landing in the sandy pathway with a distinctly firm thud.

Daughter #2 is returning to school on Monday for her mock exams week: so it seemed fair, to ask her problems about falling coconuts. True to form, daughter #2 moaned at her father, but then was interested in the discussion.

So, at what speed does a coconut fall at coconut corner? V^2=u^2+2as, where u, the initial velocity is 0, and a, acceleration is g, gravity. We can ignore drag and buoyancy. So, the square of the thudding (final) velocity is v^2=2*g*6=120, or approximately 11m/s [again, note of sadness- it is a rare colleague who can estimate root 120, which should be easy, being a bit less than root 121, which number is eleven squared].

The thud speed from a lesser, 3m, tree, is v^2=2*g*3, or somewhat less than 8 m/s.

Let us say that a coconut hit a less than happy holidaymaker. By what proportion does he wish it were not from coconut corner? Firstly, I am ignoring the 1m or so height of the person, so the speeds are less, but note in passing that life isn't fair, and the child would be hit harder than the parent. I think the hurt would be a matter of the energy in the coconut, the kinetic energy, given by 1/2 mv^2, so that the coconut corner one would hurt twice as much.

I am not sure though that I am right, and this is where my lack of deep understanding of force and energy come into play; and my lack of understanding, fortunately, of falling coconuts. But I think the coconut's kinetic energy is dissipated as sound (the thud, not the ow!), probably kinetic energy- unless the coconut hits mid page, I presume the holidaymaker is toppled a bit, if not knocked off their feet; probably kinetic energy into the ground too, and also perhaps into heat-the head will get somewhat hotter. Against that supposition, the coconut must come to rest, so must decelerate: to the extent it bounces off I am not sure what happens, but simplifying to it coming to rest atop, then the head exerts a breaking force ma, and the coconut exerts an equivalent force on the head. I suspect, but am guessing, that here we are dealing with the linear velocity, not its square, so 11/8, or, more precisely, root 2: 1.4 times.

Enough on this, otherwise I risk going coconuts. Of course, the questions as posed are ridiculously but are essentially the same as numerous practical issues, such as the design of cycle helmets, to name but one. I doubt I will have time to research it once back at home, but maybe one of my readers has a better understanding and can share his or her views?

 

From → General, Maths, Travel

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