Determining g by free fall with light gates -- assumption analysis

[maxim07]

Homework Statement

A method of determining g by free fall is to use a light gate to record the time it takes for a card of length l to fall through it after being dropped from a distance s. The velocity of the card is calculated using v = l/t. A graph of v^2 vs s will have a gradient 2a. Where a is g. One assumption that has to be made is that the light gate is measuring the average velocity because the card is still accelerating when falling through the gate. Because of this the distance s that the card is dropped from is measured from the cards midpoint, as this is the velocity that the light gate will record. But this assumes that the acceleration is constant but air resistance means it is not constant, so the mean velocity may not actually be at the midpoint of the card.

Is there any way that this can be corrected?

Relevant Equations

v = l/t
v^2 = u^2 + 2as

the only way i can think of changing the method is by using 2 light gates instead of 1, calculating the velocity through each and using this to get an average velocity, but not sure how this would help

 

[haruspex]

I can think of several ways...
Run the experiment again, in a partial vacuum.
Take multiple measurements with different drop distances.
Take a measurement with a double thickness of card.
I'll leave you to think about how you would use the extra measurements to refine the result.

 

[maxim07]

I should have included more detail about the experiment. I think the experiment is supposed to be basic and so that we consider its limitations and the errors involved, so I don’t think a partial vacuum is possible. blu tack is used to weigh down the card, so we don’t need to increase the thickness and the card is dropped from multiple heights so that a graph of v^2 against s can be plotted. The gradient is used to determine a which reduces random error. Parallax error is avoided when measuring s. As mentioned s is measured from the light beam to the midpoint of the card.

 

[haruspex]

blu tack is used to weigh down the card, so we don’t need to increase the thickness and the card is dropped from multiple heights so that a graph of v^2 against s can be plotted.

I was thinking of running it with different weights of card but which should have the same drag.
The trouble with varying the weight and/or the drop length and plotting the results is that you may have to make an assumption about how drag varies with speed.

 

[maxim07]

if the cards have the same drag this would not reduce the air resistance, so how would this help?

 

[Steve4Physics]

if the cards have the same drag this would not reduce the air resistance, so how would this help?

Drag force depends on a object's speed. What difference in your results would you expect between using a sheet of thick cardboard and a sheet of lead of the same size? (And why?!)

 

[maxim07]

The drag force is proportional to the square of the object‘s speed but also dependent on area in constant with air. Since both sheets are same size air resistance is only dependent on the velocity. The lead sheet experiences a greater gravitational force so would it accelerate quicker and have a greater velocity, but then wouldn’t the drag force on it be greater too since drag is proportional to v^2?

 

[haruspex]

The lead sheet experiences a greater gravitational force so would it accelerate quicker

That does not follow quite so simply. It also has greater mass. It falls faster because, at a given speed, the drag is a smaller in relation to the gravitational force.

then wouldn’t the drag force on it be greater too since drag is proportional to v^2?

Yes, but it's still going to be a bit faster.

But what you need to do in this thread is post some equations. Let's start off without any assumption of how drag depends on speed, just writing it as F_d(v).
Write the equation, and see if there's anything can be said about having values for v with different masses.

 

[maxim07]

So if we don’t assume any relationship should I just write it as a normal force in the form F = ma but as a function of v so a = δv/t. If the objects are dropped from rest u = 0 so a = v/t

F_d(v) = mg - mv/t

So if the objects have the same velocity the drag force on the heavier object would be greater. But this is the final drag force as the drag force for a heavier object has to be greater than a lighter object for it to reach terminal velocity. Or am I supposed to see what happens with different velocities ?

 

[haruspex]

F_d(v) = mg - mv/t

No, that should be dv/dt, not v/t.
I was hoping to find a way we could make progress without any assumption about F_d, other than being monotonically increasing. But I don't see any.
The trouble is that at low speeds it will be about linear, but at higher speeds quadratic.

 

[maxim07]

That does not follow quite so simply. It also has greater mass. It falls faster because, at a given speed, the drag is a smaller in relation to the gravitational force.

so at small speeds the drag increases linearly with velocity and the velocities in this experiment are small, so using an object with a greater mass would reduce the effects of air resistance and make the recorded mean velocity closer to what it would be if it was accelerating at g.

 

[haruspex]

at small speeds the drag increases linearly with velocity

Yes.

velocities in this experiment are small

Not sure it's obvious that they're small enough to make it linear. Look up Reynolds' number.

using an object with a greater mass would reduce the effects of air resistance and make the recorded mean velocity closer to what it would be if it was accelerating at g.

Yes, but that is true whether the drag is linear with velocity, quadratic or whatever.

 

[maxim07]

the object in this experiment is dropped from heights ≤ 1m. So the maximum velocity would be 4.43 m s^-1 if the object is dropped from rest if it was in free fall. So can I assume that using a card of greater mass would make the recorded velocity closer to this as drag would be reduced. If it is falling at a greater velocity through the light gate I’m guessing that the uncertainty in time would be increased since a smaller time interval will be recorded.

 

[haruspex]

the object in this experiment is dropped from heights ≤ 1m. So the maximum velocity would be 4.43 m s^-1 if the object is dropped from rest if it was in free fall. So can I assume that using a card of greater mass would make the recorded velocity closer to this as drag would be reduced. If it is falling at a greater velocity through the light gate I’m guessing that the uncertainty in time would be increased since a smaller time interval will be recorded.

Yes.
Depending on how much time you have to devote to it, and just how accurate you need to get, you could

• Use several different masses,
• See if the results fit linear or quadratic drag or whatever,
• Extrapolate to zero drag

 

[Steve4Physics]

Homework Statement:: A method of determining g by free fall is to use a light gate to record the time it takes for a card of length l to fall through it after being dropped from a distance s. The velocity of the card is calculated using v = l/t.

Hopefully the card will not rotate while falling. If it does, its 'effective length' will be longer than the true length (l).