Particle's maximum height in liquid

[fullerene]

I'm not sure if this is the appropriate subforum, but here it goes:

Suppose you have a particle, with the mass m, at the bottom of jar which is filled with liquid. The particle is shot upwards, parallel to the gravitational force, with the speed v0. The drag force is proportional to the particle's speed. What is the maximum height ymax of the particle?

First, "down"force equals to:
Ft=W+Fd
m*a=m*g+b*v
=> a(v)=g+(b/m)v

v(t)=v0+a*t, at ymax: v(t)=0
=> 0=v0+a*t
...

With cross subtituting, I cannot eliminate the variables v or t from the ymax function. Somehow I think the ymax function should not depend on the v or t variables, but maybe I'm wrong. Can you guys help me?

Thanks!

 

[MathematicalPhysicist]

i think it should be:
Ft-W-Fd=ma
where Ft is force that needed to shoot the particle.

 

[fullerene]

Hmm... I think that the force to shoot the projectile is irrelevant because the projectile has experienced the accelerating process and the force that "shoot" the projectile is no longer acting on it. I dunno, maybe I'm wrong.

Btw, when we consider that this experiment is done in vacumm, the ymax function is obtainable rather easily, just use y(t)=v0*t+(1/2)*g*(t^2). The value of t can be obtainable using v(t)=v0+g*t, which v(t) is zero (at ymax). Then ymax is just a function of v0 -> ymax(v0)=a*v0+b.

 

[MathematicalPhysicist]

i think you can use here energies:
$$K_{h}-K_{v_0}=mgh+bv^2/2$$
and the fact that: h=(v^2-v0^2)/2g (also known as galileo equation).
but I am not sure about the rhs of the first equation.

 

[HallsofIvy]

Since a= dv/dt and depends on v, you can't solve that algebrically. You have a differential equation:
[tex]\frac{dv}{dt}= -g- \frac{bv}{m}[/itex]
That's relatively simple "non-homogeneous linear differential equation with constant coefficients".

 

[fullerene]

Ok, thanks HallsofIvy and loop_quantum_gravity.