You notice a tiny piece of ?y ash that is ejected from the ?re vertically upward

Question

You notice a tiny piece of ?y ash that is ejected from the ?re vertically upward with initial speed v0. Since physicists tend to love to think about casual observations like this more formally, you decide to try to predict if the ash will reach tree-level. Let’s measure the position of the ?y ash, y from the point of release, once again taking vertically upward as
the positive direction.
(a) Clearly the physics of “?ying ash” could be very complicated! But, let’s start with the simplest possible model and go from there. If you consider ash to be a point particle of mass m acting only under gravity,
what is the equation of motion? What is the maximum height (in terms of v0 and g)?
(b) The next simplest model might be to also include simple air drag. Taylor discusses the choice between “linear” and “quadratic” drag. Knowing that ash particles are extremely tiny, and not particularly fast moving to start with, if you had to choose just ONE of these, which should it be? Brie?y (qualitatively
is all we want) justify your answer.
(c) Now, using this model, ?nd the time for the ?y ash to reach its highest point and its position ymax at that time. (Note that velocity vy(t) and position y(t) are derived in the lecture notes, and the text. Of course, it’s always a good idea to rederive them yourself, but not required for credit. However, you should be careful, because in the text, Taylor de?nes +y to be the other way, so watch your minus signs!
I suggest using notation from question 1 to de?ne symbols. No numbers here, do it all symbolically.)
(d) Show that as the drag coe!cient approaches zero your answer in part c reduces to the well known freshman physics result you got in part a. Hint: If the drag coe!cient is small the terminal velocity is big, so v0/vter is very small. Use the Taylor expansion ln(1 + epsilon)

Explanation / Answer

S=ut+0.5att Hmax=U^2/2g

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