Which of the following?

A block is projected up a rough inclined plane. It initially goes up, then comes back to the initial point. If time of ascent is t a t_a and time of descent is t d t_d . Then which of the following is correct?

None of these t a t_a < t d t_d Both come back the same time t d t_d < t a t_a

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1 solution

Steven Chase
Nov 21, 2016

The derivations for t a t_a and t d t_d are fairly straightforward. Some key points are that the gravity and friction forces add (relative to each other) on the way up and subtract on the way down. The travel distances are the same both ways. On the way down, the particle starts with no speed (unlike the upward case), but has a net force accelerating it rather than retarding it. The results are:

t a = v 0 g s i n θ + μ g c o s θ t_a = \frac{v_0}{gsin{\theta} + \mu g cos{\theta}}

t d = v 0 ( g s i n θ + μ g c o s θ ) ( g s i n θ μ g c o s θ ) t_d = \frac{v_0}{\sqrt{(gsin{\theta} + \mu g cos{\theta})(gsin{\theta} - \mu g cos{\theta})}}

By inspection, t d t_d will generally be larger than t a t_a

What would be the scenario if the plane is frictionless?

Kaushik Chandra - 4 years, 5 months ago

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They would be equal in that case. It would be essentially the same thing as throwing a ball straight up in the air (just with a coordinate transformation).

Steven Chase - 4 years, 5 months ago

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Thank you very much!

Kaushik Chandra - 4 years, 4 months ago

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