How to Derive Generation Equation for Speed of Falling Object

Question

At school we are doing an experiment with a setup involving a vertically positioned ruler, a marble and a light gate. The method requires us to drop the marble from various heights on a ruler (in 4cm increments) into the light gate and record the resulting speed of the falling marble in m/s. (Light gate was positioned at the 0cm mark). In this experiment, the weight of the marble was 5.17g. (Recorded data and corresponding graph attached).
We were then asked to form a generic equation for the speed of a falling object from a given height (using the variables 's' - for speed of falling object (m/s), 'h' - for height of object release (cm) and 'w' - for weight of object (g)).
However, I have had difficulty deriving an equation which could be used generically, since it could be assumed that a heavier/lighter object would have a different size accordingly. Thus, this would directly affect the speed of the falling object due to air resistance.
I have generated an equation from the recorded data using Microsoft Excel. However, I am not sure if this would be suitable for use generically. In addition it does not consider 'w' - weight of the object.
Do you have any suggestions as to how I could improve the current equation to make it more suitable for generic use? Or is there a certain equation which could be used for this?
Thanks.

Yejus · Answer

It looks like you're using a linear relationship between $v$ and $h$, which is incorrect. Assuming you are releasing the marble from rest, the final speed should depend on the height it was released from like so: $v = sqrt{2 g h},$ where $g approx 9.81$ $text{m} , text{s}^{-2}$ is the gravitational acceleration that does not depend on the weight of the marble. Hence, the relationship between $v$ and $h$ is not linear like you have assumed, but rather square-root. If Excel automatically fit a line for you, I suspect that was because you do not have enough measurements, so it thought a line was a good enough fit.
Also, I think air resistance should be negligible for an experiment of your scale. Try fitting your data points again using the correct relation and include more measurements if possible.

Not_Einstein · Answer

I think you should be able to safely ignore air resistance. You can use conservation of energy. The object starts out with potential energy $mgh$, where h is the height above where the speed is measured. When it reaches the point where it's measured, it has $(1/2)mv^2$ kinetic energy. Equating the two, we get $v^2 = 2gh$. So the relationship is not a straight line as your data shows, although it is close over this small range of h values.

Answered by Not_Einstein on October 28, 2021

How to Derive Generation Equation for Speed of Falling Object

2 Answers

Add your own answers!

Ask a Question