What is the Terminal Velocity of an Object Subjected to Drag Force?

[Yapper]

Homework Statement

An object of 0.5kg is dropped from rest at an altitude of h=500 meters. It
falls while being subjected to a drag force, defined as: f_D = 2v .
Derive explicitly: v(t), and y(t) and plot v(t) given the initial conditions; use: t between 0
and 0.4 seconds, and v between 0 and -3 m/s
Based on your graph for v(t), what is the terminal velocity? Confirm this result
analytically (either using the limit or using Newton).

Homework Equations

a = dv/dt v = dy/dt

The Attempt at a Solution

For my equation i get 2v - mg = ma setting ma = 0 i solve for terminal velocity and get v = mg/2 = 2.45 m/s but I am having trouble explicitly deriving v(t) and y(t) because I end up with v = 9.8t - 4y which is 1 variable too many. How do I derive the equation for v with t as the only variable.

 

[ehild]

Have you learned about differential equations?

Replacing a=dv/dt, write up the equation of motion in terms of v and t. It is a first-order linear de. Solve with the initial conditions given.

ehild

 

[Yapper]

I've taken Calc I and II and I am taking Calc III right now and I never remember having to solve these. So far based on what I just been reading i get v(t) = 2.45 - 2.45e^4t with the initial condition being V(0) = 0. is that right? and if so why is the graph backwards. it seems like it should be 2.45 - 2.45e^-4t but I don't see where I made a mistake.

My work:

dv/dt - 4v = -9.8 I multiplied both sides by e^-4t to get v'e^-4t + -4e^-4t = -9.8e^-4t

so that gets me (ve^-4t)' = -9.8e^-4t

I integrate both sides then try to get v by itself and end up with:

v= 2.45 + ce^4t with v(0) = 0 c = -2.45

 

[Yapper]

I changed my frame of reference around and it worked out better using dv/dt + 4v = 9.8.

I also got y(t) = 2.45t + 6.125e^-4t is that right?

 

[ehild]

You forgot to divide g by m. And clarify, what do you mean by y.
Edit: G has to be divided, not g.
ehild

 

[Yapper]

why would I divide gravity by mass?

y is the displacement in the y direction from the point that the object began following

 

[ehild]

Well, you are right, I mixed it with weight...

If y(t) is displacement, it has to be zero at t=0.
Also, the original height is given. I think, the problem asks the height as function of t.

ehild