First order ODE question: prove vT=uv0/sqrt(u^2-v0^2) is terminal velocity.

[wes1236]

how do you show this

a mass m is projected vertically upward with an initial velocity u. the air resistance is given as kv^2. the mass returns to point of projection with velocity v0. prove terminal velocity is given by
vT=uv0/sqrt(u^2-v0^2).
Hint distances travelled upwards and downwards are same.

 

[wes1236]

need help with first order ODE

a mass m is projected vertically upward with an initial velocity u. the air resistance is given as kv^2. the mass returns to point of projection with velocity v0. prove terminal velocity is given by
vT=uv0/sqrt(u^2-v0^2).
Hint distances travelled upwards and downwards are same.

stuck once i got 2 different equation for up motion and down motion

 

[Deleted member 4993]

how do you show this

a mass m is projected vertically upward with an initial velocity u. the air resistance is given as kv^2. the mass returns to point of projection with velocity v0. prove terminal velocity is given by
vT=uv0/sqrt(u^2-v0^2).
Hint distances travelled upwards and downwards are same.

So the differential equation is (applying Newton's first law:

m(d^2x/dt^2) -k(dx/dt)^2 + mg = 0

What is the "condition" (or definition) of terminal velocity?

What do you get from that definition? (apply Newton's first law)?