Resisted motion (1 Viewer)

[cutemouse]

A particle is moving vertically downward in a medium which exerts a resistance to the motion which is proportional to the speed of the particle. The particle is released from rest at O, and at time t its position is at a distance x below O and its speed is v. If the terminal velocity is V, show that gx+Vv=Vgt.

Thanks

 

[Trebla]

What is G?

 

[Trebla]

A particle is moving vertically downward in a medium which exerts a resistance to the motion which is proportional to the speed of the particle. The particle is released from rest at O, and at time t its position is at a distance x below O and its speed is v. If the terminal velocity is V, show that gx+Vv=G+Vgt.

Thanks

Defining downwards direction as positive:
a = g - kv
You can easily show by solving differential equations that:
x = (-1/k) {v + (g/k) ln [(g - kv)/g]}
t = (1/k) ln [g/(g - kv)]
Terminal velocity occurs when a = 0, v = V so
V = g/k
Assuming it's all correct I get:
gx + Vv
= - g/k {v + (g/k) ln [(g - kv)/g]} + Vv
= - V {v + V ln [(g - kv)/g]} + Vv
= - V² ln [(g - kv)/g]
= V² ln [g/(g - kv)]
= V²kt
= V²(g/V)t
= Vgt

 

[cutemouse]

What is G?

Sorry about that.. Typo Edited original thread...