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I was given the task to try and teach some physics (mechanics and dynamics mostly) to two young students who don't have any knowledge of math analysis such as limits, derivatives, integrals but they supposely can work with vectors and also basics of trigonometry, and I have full discretionary power about the program.

About physics we already talked about:

physical quantities, measures, errors in measures and propagation, dimensional analysis, angles and basics of trigonometry, basics of vectors (what they are and why we need them with some examples, sum, difference and product with a number with some of their properties), an introduction to motion and why it is something relative (with example), frames of reference, equation of a line and what are the slope and intercept, and eventually everything they had to know about straight line motions: the concept of average speed with its sign, instantaneous speed, average acceleration, instantaneous acceleration and derived their equations in various situations.

Now I thought that in order to fully understand the first law of motion and why the description is easier in inertial frames of reference, or even to show that what we found for straight line motion is more general and applies to any one dimensional motion, I should talk them about circular motion and I got myself an idea of how to explain them the fact that velocity is tangential to the trajectory and that $$\vec{v} = R\omega{\widehat{u}_{\theta}}$$ Where $R$ is the radius of the circle. Now I can fully show them that there must be acceleration to have a circular motion because we defined when talking about straight line motions the acceleration as the rate of change of speed (meant as the magnitude of velocity), and now the velocity vector can change not only in magnitude but also in direction, and when generalizing the acceleration as the rate of change of the velocity we need to keep that in mind. However now I'm having hard times thinking about how I could derive them the fact that acceleration vector must have a tangential component due to the change of magnitude and a radial acceleration pointing toward the center due to the change in direction. When I studied that I obtained it as a math and geometry exercise involving derivatives (I studied that at computer engineering university)

Lately I thought that I could even try to simulate the definition of derivative and for the radial acceleration ask them what would be the direction if the speed didn't change (of course it'd be pointing toward the center) and then for the tangential one asking them what would be the direction of the acceleration if only the speed changed and not the direction of velocity (of course tangential), but what about their expressions?

Qmechanic
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Baffo rasta
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  • You can easily show this by the famous eraser on a string experiment. – infinitezero Jul 21 '19 at 11:31
  • @infinitezero: it could be interesting, I never heard about this experiment, could you please provide some reference? Thanks! – Baffo rasta Jul 21 '19 at 11:34
  • Just tie an eraser to a string and let the students rotate it. They can feel the force they need to apply on the strong to keep it rotating. The string is obviously tangential and since $F=ma$ you've shown that there is indeed acceleration in that direction – infinitezero Jul 21 '19 at 12:38
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    There is a well known construct used in many algebra/trig based textbooks. I give a brief version of it in an earlier answer to a question which is a good candidate for a duplicate. – dmckee --- ex-moderator kitten Jul 21 '19 at 14:05
  • @dmckee Thanks for posting that link, I really like how the problem is being solved for the centripetal acceleration. Any suggestion about the tangential one due to the change of speed? – Baffo rasta Jul 21 '19 at 14:15
  • @infinitezero The string isn’t tangential. – G. Smith Jul 21 '19 at 18:02
  • If course you're right, the string is radial, as the force is. I just had the word tangential from OP in mind – infinitezero Jul 21 '19 at 18:46

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