Deceleration of a High-Speed Projectile in Water

Introduction

The high-speed video clip to the right shows the descent of a spherical projectile (BB) into a tank of water. The BB hit the water at 228 m/s (Mach 0.67). Successive frames are 0.000300 s (300 µs) apart. As the BB descended in the tank, it pushed the water aside, producing a cavity behind it. The expansion of the cavity's walls was rapid at first but then slowed, came to a stop and collapsed inward. The complete collapse of the cavity is not shown.

The speed and extent of the expansion depend on the speed of the projectile, which decreases due to the upward drag force of the water. The purpose of this lab is not to investigate the cavity, interesting as that may be. The goal is to measure the position of the BB as a function of time in the water and compare to theoretical predictions.

A note about how the movie was made: The clip was produced by extracting each image from the original multiple-image photograph that contained all twelve images. That photograph, which was taken in collaboration with Taylor Hinshaw, is shown below. Successive images of the BB's path were swept horizontally by a rotating mirror that reflected the images to the camera. Therefore, the horizontal dimension is a time scale rather than a distance scale. This technique was used so that the images would not overlap and obscure useful information about the collision.

The black areas of the photograph are either air or calm water. White areas show disturbances in the water, much as one would see in a turbulent stream. The interior cavity walls, for example, appear white where air and water mix. The surface of the water at the top of the tank is indistinct due to reflections from the under surface of the water. (A horizontal line has been added to the first frame of the movie to indicate the water level.)

Theory

Review the topics of Reynold's number, drag coefficient, and the distinction between laminar and turbulent flow. Why we can use a drag force proportional to v² for the situation of this lab?

The Problem

You'll obtain position versus time data from the video clip. You'll then fit the data to your theoretical equation and deduce the value of the drag coefficient from the coefficients of the fit. A problem that you'll need to address is that we don't know for sure when t = 0 occurs. The fit is sensitive to this initial condition.

Analysis

Here's some information and hints for your video analysis:

1. The position of the BB is not well-defined but we know that it's at the vertex of the conical cavity. The diameter of the BB is about the same as the width of the cavity in this area. Try to click on where you judge the center of the BB to be and then mark your points consistently throughout.
2. After you've marked all the points, display the trail of points. You'll probably notice some horizontal deviations of the points from the vertical. This results from registration errors in extracting images from the photograph in order to produce the video clip. You won't need the x-coordinates in your analysis, so just ignore them. The y-coordinates, on the other hand, are important. Any registration error here would be no more than a pixel.
3. For scaling purposes, the first image is at a depth of 0.0375 m and the last image is at a depth of 0.2680 m.
4. Translate the axes so that the x-axis coincides with the water line shown on the first frame. Then rotate the axes 180° so that +y is down.

Do the following in your graphical analysis.

1. Create a corrected time column. The movie clip runs at 30 frames per second, but the actual time between images is 0.000300 s (or the equivalent of 3333 fps).

2. Fit your y vs t (corrected) data to the theoretical equation. The fit equation you need isn't in the list of stock functions. However, you can simply type the function into the box beside y = . Use A and B to represent your constants and x to represent the time.

3. You should find that the fit is poor, especially near t = 0. This is probably due to the fact that the video analysis assumed that the first image occurred at t = 0. Yet it's obvious from the video clip that the actual time at which the BB entered the water was earlier. Let's define a third constant, C, to use in your fit equation. This constant will be the time offset, that is, the time interval from the BB's breaking the water's surface to the first image. Redo your fit, but this time, add C to x. When you do the fit, expect to get much better results.

4. Find the residuals of your fit.

Interpretations

1. Write the equation of the fit, substituting actual variable symbols for the generic x and y that Graphical Analysis gives.  Include the numerical values of the fit coefficients with units and correct significant figures.  Below the equation of fit, write the theoretical equation for position of the BB below the water as a function of time.  

2. Take the derivative of the equation of fit.  By comparing to theory, deduce the value of the velocity at t = 0.  Compare to the value of 228 m/s that was given in the introduction of the lab.   (This value was measured by timing the passage of the BB between two photogates, 100 mm apart, just beyond the end of the rifle barrel.)

3. Ask for a container of BBs so that you can measure the mass and diameter for yourself.  Tell what measuring instruments you used.  Describe any special techniques of measurement.  Record the values of mass and diameter.  Also record your estimate of the accuracy of each measurement.  This estimate should be based on a) the precision of the measurement instrument, and b) the method of measurement.

4. From the equation of fit and the theoretical equation for position as a function of time, deduce the value of the drag coefficient, C_d.  Show your method, starting with relevant equations, solving for C_d, substituting known values with units, and reducing.

5. Compare the value of C_d obtained in 4) with the value obtained in the prelab using the graph of C_d vs. R.

6. How large is the time offset? Does the value make sense?  Explain.

Discussion and Conclusion

Summarize your results and account as best you can for the percentage differences found above.  Include a discussion of the limitations of assuming a v² drag force to describe the experimental situation.  Don't dwell on factors that we know are insignificant (weight and buoyancy).