Question

Sam, whose mass is $75kg$, straps on his skis and starts down a $50$-m-high, ${20}^o$frictionless slope. A strong headwind exerts a horizontal force of $200$N on him as he skies. Use work and energy to find Sam’s speed at the bottom.

Short answer

Sam's speed at the bottom of the slope is$15.76m/s.$

Step-by-step solution

Given information

Sam, whose mass is$75kg$straps on his skis and starts down a$50m$high,${20}^o$frictionless slope. A strong head wind exerts a horizontal force of$200N$on him as he skies.

Step 2:Explnation

Potential energy of a mass at height h is taken as mgh. KE of an object moving with velocity $vis\frac{1}{2}m{v}^2.$

The component along the slope of the wind force acting on Sam is $200\cos \left(\theta \right)=200\cos \left({20}^o\right).$

Work done by the wind on Sam is

$$\begin{gathered} W=Force\times dis\tan ce \\ W=200\cos \left({20}^o\right)\times hcsc\left({20}^o\right) \\ W=200cot\left({20}^o\right)\times 50 \\ W=27,474.77J \end{gathered}$$

The KE of Sam at the bottom is

$$\begin{gathered} \frac{1}{2}m{v}^2=mgh-W \\ {v}^2=\frac{2\left(mgh-W\right)}{m} \\ {v}^2=\frac{2\left(75\times 9.81\times 50-27,474.77\right)}{75} \\ {v}^2=248.34 \\ v=15.76m/s \end{gathered}$$

The other component (component perpendicular to the slope) of the wind force does no work as the friction between the skis and slope is zero.