Linear model of air resistance

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Homework Statement


Hi,
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.

Model:
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
x= displacement
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.

Homework Equations



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:

x= [mg/k]*(t+(m/k)*[e^(-kt/m)-1])

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.
 

Answers and Replies

  • #2
haruspex
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Your experimental result has h = ct+d. So for large t, h ~ ct. Doesn't that give you your terminal velocity? And presumably you know m.
 

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