Question

A ski that is placed on snow will stick to the snow. However, when the ski is moved along the snow, the rubbing warms and partially melts the snow, reducing the coefficient of kinetic friction and promoting sliding. Waxing the ski makes it water repellent and reduces friction with the resulting layer of water. A magazine reports that a new type of plastic ski is especially water repellent and that, on a gentle 200 mslope in the Alps, a skier reduced his top-to-bottom time from 61 swith standard skis to 42 swith the new skis. Determine the magnitude of his average acceleration with (a) the standard skis and (b) the new skis. Assuming a $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{°}$slope, compute the coefficient of kinetic friction for (c) the standard skis and (d) the new skis.

Short answer

$$\begin{gathered} \left(a\right)a_1=0.11m/s^2 \\ \left(b\right)a_2=0.23m/s^2 \\ \left(c\right){\mu }_{k1}=0.041 \\ \left(d\right){\mu }_{k2}=0.029 \end{gathered}$$

Step-by-step solution

Given

New ski is 200 m slope in the Alps, a skier reduced his top-to-bottom time from 61 s with standard skis to 42 s .

Understanding the concept

Frictional force is given by the product of coefficient of friction and the normal reaction. Newton’s 2nd law states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the object’s mass. Using these concepts, the problem can be solved.

Formula:

${\mathbf{F}}_{\mathbf{net}}\mathbf{=}\mathbf{mg}\mathbf{}\mathbf{sin\theta }\mathbf{-}{\mathbf{f}}_{\mathbf{k}}$

 Calculation for average acceleration with the standard skis

(a)

If the distance is L during time t with initial speed zero and a constant acceleration a, then

$L=a{t}^2/2,$

Which gives the acceleration for the first (old) pair of skis:

$$\begin{gathered} a_1=\frac{2L}{t_1^2} \\ =\frac{2\left(200m\right)}{{\left(61s\right)}^2} \\ =0.11m/s^2 \end{gathered}$$

 Calculation for average acceleration with the new skis

(b)

The acceleration for the second (new) pair is

$$\begin{gathered} a_2=\frac{2L}{t_1^2} \\ =\frac{2\left(200m\right)}{{\left(42s\right)}^2} \\ =0.23m/s^2 \end{gathered}$$

Step 5: Assuming a 3.0° slope, compute the coefficient of kinetic friction for the standard skis

(c)

The net force along the slope acting on the skier of mass m is

$$\begin{gathered} F_{net}=mg\sin \theta -f_k \\ =mg\left(\sin \theta -{\mu }_k\cos \theta \right) \\ =ma \end{gathered}$$

Which we solve for for the first pair of skis:

$$\begin{gathered} {\mu }_{k1}=\tan \theta -\frac{a_1}{g\cos \theta } \\ =\tan \left(3°\right)-\frac{0.11m/s^2}{\left(9.8m/s^2\right)\cos \left(3°\right)} \\ =0.041 \end{gathered}$$

Assuming a 3.0° slope, compute the coefficient of kinetic friction for the new skis

(d)

For second pair, we have

$$\begin{gathered} {\mu }_{k2}=\tan \theta -\frac{a_2}{g\cos \theta } \\ =\tan \left(3°\right)-\frac{0.23m/s^2}{\left(9.8m/s^2\right)\cos \left(3°\right)} \\ =0.029 \end{gathered}$$