Ball sinking in water - apparent depth measurement · CAP 2004 #P1

Physics · mechanics

Problem

A uniform ball of mass M=1.00M = 1.00 kg is released from rest at the surface of a pond and sinks vertically into the water. An observer, wishing to determine the density of this ball, measures the time-dependence of the angle θ\theta between the vertical at his position, a distance hh above the surface, and the direction of the ball's apparent instantaneous position (see figure). The horizontal distance between the observer and the initial position of the ball is dd. [Figure: Shows observer above water surface at height h and distance d from ball, measuring angle θ to apparent position of sinking ball] The drag force acting on the ball as it sinks is well approximated at low speed vv by Fdrag=bvF_{drag} = bv, where bb is a constant. We also take the density of water ρw\rho_w to be ρw=1000\rho_w = 1000 kg/m³ and its refraction index as nw=1.33n_w = 1.33. (a) Derive an expression for the instantaneous depth yy of the ball as a function of θ\theta, hh, and dd. (b) Having calculated the depth of the ball as a function of time, the observer can then construct the following table for the speed of the ball as a function of time:
Time [s]Speed [m/s]Time [s]Speed [m/s]
0.0000.0000.2250.8425
0.0100.09100.2750.8767
0.0250.20860.3250.8966
0.0750.50810.3750.9082
0.1250.68230.4250.9145
0.1750.78360.4750.9155
From these data, find the initial acceleration of the ball and estimate at which value its speed will stabilise. Then calculate the density of the ball and obtain a value for the drag coefficient bb.

Browse the problem library