Question

I am doing a physics lab report, but I need some help with the math involved. The lab involves dropping different numbers of coffee filters (n) off a balcony and recording the time of descent. We're talking about terminal velocity and drag forces.

Task 1: Derive a general equation for the velocity v^n of a stack of n coffee filters each of mass m falling through the air after terminal velocity has been achieved.
I really need help understanding what to do. We know that F = bv + cv^2, and that for a slow moving object such as a coffee filter falls, the cv^2 term is effectively 0.

Answer 1

I have more information on what we have to do for this lab, there are 2 other tasks, but as the question box only has so much room I'll wait until they're needed.

Answer 2

Help is sincerely appreciated if you know how to do it!

Answer 3

*** velocity v^n of a stack of n coffee filters ***
what is v^n represent? do you mean v(n) (velocity as a function of n) ?

Answer 4

at terminal velocity, the force due to gravity is balanced by the force due to resistance.

Fg - Fr = 0 or
Fg= Fr

Fg= Ma = n*m*g
Fr= bv + c v^2
if we equate these forces we get
\[ n m g= bv +c v^2 \]
where v is the terminal velocity
\[ c v^2 + b v -n m g = 0 \]
using the quadratic formula we can solve for v
\[ v = \frac{1}{2c} \left( -b \pm \sqrt{b^2 +4nmcg}\right) \]

Answer 5

I follow all of the math, but where does that n come from? I think it's velocity raised to the n power, and we're asked to solve for v^n. Especially because The second part of the lab has us do this:

Answer 6

"Applying logarithms to the equation for velocity you derive in Task 1, manipulate it to develop an equation in which the power of the velocity, n, can be related to the slope of a best-fit straight line fit to the experimental data."

Answer 7

*** We know that F = bv + cv^2 ***
is this the force due to air resistance ?

Answer 8

Yes. It's the drag force which for this case is air resistance. But in all our class examples, we've canceled the cv^2 term because it becomes very nearly 0 for slow moving objects.

Answer 9

So for this, I think it's F = bv.
But I think, based on the rest of the lab that the v in that equation is v^n, so the same process applies just solving for that instead.

Answer 10

Do you have any more detail on your resistance formula?
are b or c functions of mass? does the mass of the object affect the air resistance ?

Answer 11

That was my other question, I have no idea what b and c are.

Answer 12

except that I know b is measured kg/s.

Answer 13

what shape are the filters? cones?

Answer 14

You know the white thin ones that go in coffee pots? They are sort of like cones with points cut off.

Answer 15

*** except that I know b is measured kg/s.***
are you sure it's not kg/m
that would make more sense:

F is in newtons = kg * m/s^2

to get newtons from
b * v^2
kg/m * m^2/s^2 = kg * m/s^2

Answer 16

Based on v=(ma)/b it should be kg/s.

Answer 17

oh. I was using v^2 and it's v. ok

when you say
**the slope of a best-fit straight line fit to the experimental data.**

what is the experimental data?

Answer 18

also, you say
The lab involves dropping different numbers of coffee filters (n)
which means n is the number of filters.

and apparently v^n is velocity raised to the number of filters?

Answer 19

or are they saying:
let's model the force as
\[ F_r = b \cdot v^a\]
where both a and b are unknown coefficients
i.e. a is not to be confused with the number of filters.

if that is what is going on, we have
\[ v^a = \frac{n m g}{b} \]
and
\[ a \log(v) = \log\left( \frac{n m g}{b} \right) \\
a= \frac{\log\left( \frac{n m g}{b} \right) }{\log(v) }
\]

Answer 20

if the experimental data is x= log(v) and y= log(n)
where n is the number of filters and v is the terminal velocity
we might see a line
y = m x + b0 (b0 not to be confused with the "b" used in the resistance formula)
using
\[ a \log(v) = \log\left( \frac{n m g}{b} \right) \\
a \log(v) = \log(n) + \log\left( \frac{ m g}{b} \right) \\
\log(n) =a \log(v)-\log\left( \frac{ m g}{b} \right)
\]
which matches up with
log(n) = m log(v) + b0
and a is the slope