I have to model the velocity and displacement of different paper shapes assuming that air resistance, R is proportional to velocity, v.
I have also conducted an experiment, and the equation of the line
h = 1.532t - 0.366
Where h is the height the paper shape was dropped from, if t is the time taken for it to reach the ground.
R = kv
Assuming that the only forces acting on the cup are it's mg downwards and R upwards.
And that the motion is vertical only.
when t=0, v=0 and x=0
I have found the equations for v and x in terms of t. The problem I have is finding k.
k= constant in the the assumption R is prop to v.
mg= weight of paper cup
R= air resistance
dv/dt = acceleration
v= velocity of the paper cup
Downwards is taken as positive.
I will go through what I have so you know what I am working with and the nature of the problem, but ultimately the problem I have with is k.
N[II] gives : m(dv/dt) = mg -kv
(dv/dt) + (k/m)*v = g
Using an integrating factor: I = e^(kt/m)
So d(v*e^[kt/m])/dt = ∫ge^(kt/m)dt
Finishing this and using the initial conditions:
v = [mg/k]*(1-e^[-kt/m])
Integrating this we get displacement:
x = [mg/k]*(t+(m/k)e^(-kt/m)+c)
Using initial conditions:
The Attempt at a Solution
I know that t -> ∞ v-> mg/k [which will be an asymptote when plotted]
So the terminal velocity = mg/k
I also know that the x against t graph will look like a curve with an increasing gradient, until the time at which the terminal velocity is reached. At this time the gradient will become constant.
Ok so now we find k--- how?
Thanks in advance.