A Fluid Dynamics Case Study

BY: Alex Liu

About the Project

This project is meant to explore a how objects move in water. By using a number of formulas, one should be able to estimate the velocity of the ball at any given moment in this video.

Conditions of the case study

The case study involves a bit of context for the data:

The water is stationary and is not flowing constantly like ocean waves
We are not dropping the water above the water in order to minimize the influence the initial impact has over the data. As such, the ball is being dropped after already being fully submerged in water.

Forces Acting on the Rubber Ball

Force of Gravity

The Force of Gravity is defined by:

mass*gravity.

Force of Buoyancy

The Force of Buoyancy is defined by:

g(V*P_w)

Gravity*(Volume of ball * Density of water)

Force of Drag

The Force of Drag is defined by:

1/2(C*P_w*A*v^2)

1/2(drag coefficient * density of water * cross section area *velocity^2)

Note: The Orientation of the Y-Axis is negative. So movement in the negative y direction gives us positive values

Derivative of Equations

Any falling object has two phases of velocity. For most of the video, the ball is falling at terminal velocity, so it has reached a constant speed where resistances matches the force of gravity. However, we also need to account for the speed of the ball as it accelerates to terminal velocity. We can solve for it with Newton's 2nd law and other velocity derivatives shown below.

Terminal Velocity Derivative:

When Fnet = 0

Fg= Fb +Fd

mg = g(V*P_w) + 1/2(C*P_w*A*v^2)

Velocity: sqrt(2(mg-g(V*P_w))/C*P_w*A)

Velocity Before Reaching Terminal Velocity Derivative:

V = V(old) + acceleration*change in time (need to solve acceleration as the constant variable in this updating equation)

Fnet = ma

Fg - Fb - Fd = ma

mg - g(V*P_w) - 1/2(C*P_w*A*v^2) = ma

Acceleration = g - (gV*P_w)/m) - (C*P_w*A*v^2/2m)

Velocity = Vold + (g - (gV*P_w)/m) - (C*P_w*A*v^2/2m) * change in time)

Variables and their Values

Change in Time: .01 second

Density: 1000 kg/m

Mass (of the ball): .01125 kg

Cross Sectional Area (of the ball): .0004908 m^2

Drag Coefficient: .47

Gravity: 9.81 m/s

Volume (of the ball): .00000818 m^3

rESULTS:

Using the variables' values and the derivatives stated above, we can use programs like Excel to create our theoretical set of data.

Physics Graphs.xlsx

Theoretical Time vs Velocity GRaph

ACtual Time vs Velocity Graph

Note: The signs are inverted due to the axis setup in the video tracker.

Sources of errors

I admit that the drag coefficient used in this case study is not perfect. I don't have the means to calculate it, so the source for the drag coefficient was found online here: https://en.wikipedia.org/wiki/Drag_coefficient.
Refraction of light is a problem when shooting videos causing slight distortions of the image. While not significant, it could lead to slight deviations when tracking the object.

Conclusion

We can safely say that this method of finding the velocity for a falling object in water works quite well. There is not a large percent error, and these equations can be utilized for any object. The most unpredictable part of this case study is the drag coefficient which may be impossible to find without online sources. However, assuming that you have the drag coefficient, it is possible to measure the velocity of an object underwater at any given point.

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