In the sport of curling, participants take turns sliding near-cylindrical stones across an ice court towards a target, trying to get their stones as close to the target as possible after using a set of stones. A vertical cross section of a stone is depicted below, showing that the stone is in contact with the ice on a thin ring of radius $r$. The full radius of the stone is $R$, the mass of the stone is $m$ and the coefficient of friction with the ice is $\mu$.

Consider the case when the stone is released at a speed $v_{0}$ with the aim of knocking out an opponent's stone at a distance $s$.

i  ^1.00 Give an expression for the sliding speed $v_{s}$ as a function of the time $t$ since the stone was released until the stone hits the opponent's stone.

ii  ^1.00 What is the sliding speed $v_{\text {hit }}$ just before the stone hits the opponent's stone?

Now the stone is given a small rotation at the initial angular speed $\omega_{0}$ (this can be done to alter the deflection angle of the stone when it hits the opponent's stone). Assume that the rotation speed $\omega$ remains small throughout the sliding motion: $\omega r \ll v_{s}$. Keep only the main non-vanishing terms in your calculations, i.e. among the terms with a factor $\left(\omega r / v_{s}\right)^{n}$, keep only the term with the smallest $n$. Depending on your approach you may use the following approximations for $x \ll 1$ : $(1+x)^{\alpha} \approx 1+\alpha x+\frac{1}{2} \alpha(\alpha-1) x^{2}, \sin (\alpha+x) \approx$ $\sin \alpha+x \cos \alpha, \cos x \approx 1-x^{2} / 2$. You may also need the integral $\int(a t+b)^{-1} d t=a^{-1} \ln \mid a t+$ $b \mid+C$.

iii  ^2.00 By how much does the friction force on the stone change due to its rotation? Express your answer in terms of the current angular speed $\omega$ and sliding speed $v_{s}$.

iv  ^2.00 Give an expression for the torque $\tau$ exerted on the stone.

v  ^2.00 What is the angular speed of the stone just before it hits the opponent's stone?