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Mechanics: Vertical projection with air resistance

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A particle is projected vertically upward in a constant gravitational field with an initial speed of v_0. Show that if there is a retarding force proportional to the square of the instantaneous speed, the speed of the particle when it returns to the initial position is:

(v_0*v_t)/[(v_0)^2 + (v_t)^2]^1/2

where v_t is the terminal speed.

Please show the frame of reference used and all steps.

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Please refer to the attachment for complete solution.

As the particle travels upwards, it is subjected to downward forces weight mg and air drag fd where fd = kv2 where v is the instantaneous speed and k is the proportionality constant.

Net force on the particle F = mg + kv2

Or instantaneous acceleration a = (mg + kv2)/m .......(1)

Let the particle move up by a small distance dy in time dt and its velocity decreases from v to (v-dv)

Applying v2 - u2 = 2as to the particle for its displacement dy :

(v - dv)2 - v2 = 2[(mg + kv2)/m] dy

Or v2 + dv2 - 2v dv - v2 = 2[(mg + kv2)/m] dy

Ignoring dv2 as negligible we get : - v dv = [(mg + kv2)/m] dy

Or dy = - (m/k) [v/[(gm/k + v2)] dv

Or y = - (m/k) ∫[v/[(gm/k + v2)] dv + C where C is the constant of ...

Solution Summary

The expert examines vertical projection with air resistance in mechanics. Instantaneous speeds are analyzed.