# Problem 1 – Hint 6

Here are step-by-step instructions.
1. Calculate the added mass of a thin circular disc of radius $R$. Keep in mind that this will describe the momentum of water only during a very short time period when the cylinder is immersed into water only very shallowly (during ca two frames in the video). Don’t forget that only the lower half-space is filled with water. Detailed suggestions about how to calculate this added math without using advanced math are given in Hint 4. To ensure there are no mistakes in your calculations, I recommend comparing your expression for the added mass with the one you found in the literature (a simple internet search will help). However, if you fail for some reason following Hint 4, you may try understanding a textbook where this added mass is calculated.
2. Measure from the video $v_0$ and $v_1$, the velocities of the cylinder immediately before and after plunging into water, in arbitrary units. Units don’t matter because we’ll be calculating the ratio of velocities from where the units cancel out. Measuring $v_0$ with a reasonably good accuracy is a fairly easy task. Indeed, Indeed, one can easily arrive to the conclusion that the frame rate is fast enough so that the speed increase due to the free fall acceleration can be neglected, even if a period covering several frames is considered. When using displacement over several frames, the relative accuracy of the measurement will increase. If you want to do even better, you can still consider even longer time periods while taking into account the effect of the free-fall acceleration, which means that instead of a linear regression of the cylinder’s displacement versus frame number data, you need to do a quadratic regression. The real challenge is determining the $v_1$ as this needs to be based on two neighboring frames. The first frame needs to show that the cylinder has already touched the water (if we were to use the previous frame, too, then during a part of the inter-frame period, the cylinder would have had the speed $v_0$). In the next frame, the cylinder immersion depth is small, i.e. the frame is usable, but that cannot be said about the following frame. Uncertainty is increased by the fact that the image of the cylinder is not exceedingly sharp, so it is difficult to determine the exact position of the cylinder. A small hint: instead of trying to figure out the contour of the cylinder, copy a small piece from the image of the cylinder in one frame, and displace it when it is overlaid to the other frame until the two images seemingly merge.
3. Apply the momentum conservation law: the momentum of water and cylinder together is conserved. The impact is essentially a plastic one, hence the energy is not conserved. What happens most likely to the energy is that during the impact, a short shock wave is generated that travels through water and carries away the excess energy. Note that the momentum carried away by the shock wave is small and can be neglected because its energy-to-momentum ratio is given by the speed of sound in water that is much larger than the speed of the cylinder.
4. To obtain the final answer, you will also need the radius-to-height ratio of the cylinder that can be easily measured from a video frame.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com.