Differential Eqn: Find an expression of the velocity of a rock.

[coolusername]

Homework Statement

A 20 kg rock falls with the force of gravity acting on it. Another force acts on the rock (same direction as gravity) that is proportional to the velocity squared, 100v^2. Come up with an expression of the rock's velocity at any time t.

Homework Equations

Hint: (d/dx)(tan^-1(x)) = 1/(1 + x^2)

The Attempt at a Solution

F(net) = ma = Fg + 100v^2

ma = mg + 100v^2
a = g +(100v^2)/m (m=20)
dv/dt = g + 5v^2
dv/dt = g((5/g)v^2 + 1)
dv/((5/g)v^2 + 1) = g(dt)
dv/((root(5/g)*v)^2 +1) = g(dt) (integrate both sides)
tan^-1(root(5/g)*v) = gt + C
root(5/g)*v = tan(gt + C)

v(t) = (root(g/5))tan(gt + C)

I want to know if I can get rid of the constant as well as if my approach to the general solution is correct. Also if my general soln is correct.

Thanks!

 

[mfb]

You cannot ignore units like that. 20 and 20kg are completely different things.
The force given as 100v^2 has mismatching units, are you sure there is nothing else given there?

You cannot get rid of the constant as you do not know the initial velocity.

 

[coolusername]

That's all that the question gives. There's no initial conditions given either. I assumed that the additional force would equal 100v^2 as it says it's proportional. But you're right, the units don't match up.. Could it be possible that the units are hidden within the coefficient of v^2?

 

[olivermsun]

Could it be possible that the units are hidden within the coefficient of v^2?

Yes, there has to be a dimensional coefficient with the proper units.

Normally the force proportional to v² would be something like the drag force, which acts in the direction opposite to the velocity, hence against gravity in the example. But oh well.