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How to Derive Generation Equation for Speed of Falling Object

Physics Asked on October 28, 2021

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.

Graph
Data
Schematic Diagram of Experimental Setup

2 Answers

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.

Answered by Yejus on October 28, 2021

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

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