How to Derive Generation Equation for Speed of Falling Object?

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 everyone for those responses. It has definitely clarified things for me. With regards to the equation mentioned, I am assuming everything is in SI units; so velocity - m/s, height - m, acceleration of gravity - ms^-2. Considering air resistance also in this experiment, would there be any way to account for it, in for example, an object with a considerably larger surface area? As I understand, larger surface area will result in increased air resistance of the falling object, and thus decreased acceleration. Jul 22, 2020 at 2:32

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.
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.