Question #104048
A particle of mass m is thrown vertically upward with velocity v0. The air resistance is mg cv^2 where c is a constant and v is the velocity at any time t. Show that the time taken by
the particle to reach the highest point is given by v0 underroot c= tan (gt underroot c )
1
Expert's answer
2020-03-03T15:06:32-0500

A particle thrown vertically with air resistance.

Initial speed of a particle: v0v_0 (up). 

Air resistance modeled as Fair=cv2,F_{air}=-cv^2, where cc is a constant and vv is the velocity at any time t.t.

Using Newton’s 2nd law gives:


my¨=mgcy˙2m \text{\"{y}}=-mg-c\text{\.y}^2

Alternatively, one can write


dvdt=gcmv2{dv \over dt}=-g-{c \over m}v^2

dt=dvg+cmv2dt=-{dv \over g+{c \over m}v^2}

Integrate


0thdt=v00dvg+cmv2\displaystyle\int_{0}^{t_h}dt=\displaystyle\int_{v_0}^0-{dv \over g+{c \over m}v^2}

th=mcg[arctan(cmgv)]0v0t_h=-\sqrt{{m \over cg}}\bigg[\arctan\big(\sqrt{{c \over mg}}v\big)\bigg]\begin{matrix} 0 \\ v_0 \end{matrix}

th=mcgarctan(cmgv0)t_h=\sqrt{{m \over cg}}\arctan\big(\sqrt{{c \over mg}}v_0\big)


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Comments

Assignment Expert
21.03.20, 17:19

Dear simran. Please use the panel for submitting new questions.

simran
21.03.20, 11:23

Suppose a viscous oil, whose flow is in the laminar regime is to be pumped through a 10 cm diameter horizontal pipe over a distance of 15km at a rate of s/ 10 m −3 3 . Viscosity of the oil is 03.0 poise. What is the required pressure drop to maintain such a flow

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