2023 ACJC P2 Q3

The Stoke’s Law states that when a spherical object of radius \(R\) is falling at velocity \(v\) in a fluid, the resistance force \(F\) acting on the sphere is given by

\(F=-6\pi \eta Rv\),

where \(\eta \) is the coefficient of viscosity of the fluid. The following table shows the coefficients of viscosity of various fluids at \({{25}^{{}^\circ }}\mathrm{C}\):

A spherical ball of radius \(0.05\) m and mass \(0.2\) kg is released from rest in a huge container filled with corn syrup at \({{25}^{{}^\circ }}\mathrm{C}\). It can be shown that as the ball falls through the corn syrup, the velocity \(v\) in \(\mathrm{m}{{\mathrm{s}}^{-1}}\) at time \(t\) seconds after release satisfies the differential equation:

\(\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{6\pi \,\eta \,R}{m}v=g\),

where \(g=9.780\)ms\(^{-2}\) is the acceleration due to gravity and \(m\) kg is the mass of the ball.

1. Find \(v\) in terms of \(t\).[5]
2. State the velocity of the ball after a long time, assuming that it does not reach the bottom of the container.[1]
3. The distance travelled by the ball from the point it was released is denoted by \(x\) m. Using the fact that \(v=\frac{\mathrm{d}x}{\mathrm{d}t}\), find the distance covered by the ball after \(30\) seconds.[2]