Chapter 6, Problem 31P

A novice skier starting from rest, slides down an icy frictionless 8.0° incline whose vertical height is 105 m How fast is she going when she reaches the bottom?

To determine

Velocity of novice skier when she reaches bottom

Answer to Problem 31P

Solution:Velocity of novice skier when she reaches bottom is 45.36 m/s.

Explanation of Solution

Given:

  $h_i=105\text{ m}$

Formula Used:

The forces acting on the skier is the force due to gravity and normal force. Work done by the normal force is zero because normal force is perpendicular to the direction of motion.

According to law of conservation of energy

  $\begin{array}{l}{E}_{initial}=E_{final}\\ mg{h}_i+\frac{1}{2}m{v}_1{}^2=mg{h}_f+\frac{1}{2}m{v}_f{}^2\end{array}$

Where m is the mass of skier, $h_i$ is the initial height of skier, $v_i$ is the initial velocity of skier, $v_f$ is the final velocity of the skier, $h_f$ is the final height of the skier, g is acceleration due to gravity.

Calculations: Substituting all the known values in above equation and solve for final velocity:

  $\begin{array}{l}mg{h}_i+\frac{1}{2}m{v}_1{}^2=mg{h}_f+\frac{1}{2}m{v}_f{}^2\\ $ m\times 9.8\times 105+\frac{1}{2}m{(0)}^2=mg(0)+\frac{1}{2}m{v}_f{}^2$v_f{}^2=2\times 9.8\times 105\\ v_f{}^2=2058\\ v_f=45.36\text{ m/s}\end{array}$

Velocity of novice skier when she reaches bottom is $45.36\text{ m/s}.$

Conclusion:Novice skier reaches the bottom at $45.36\text{ m/s}$ when skier starting from rest, slides down a frictionless $8.0°$ incline whose vertical height is 105 m.