A skier approaches the base of an icy hill with a speed of 11.6 m/s. The hill slopes upward at 24° above the horizontal. Ignoring all friction forces, find the acceleration of this skier (a) when she is going up the hill, (b) when she has reached her highest point, and (c) after she has started sliding down the hill. In each case, start with a free-body diagram of the skier.
To find the acceleration of the skier in each case, we need to analyze the forces acting on the skier using a free-body diagram.
(a) When the skier is going up the hill:
The only force acting in the horizontal direction is the component of gravitational force parallel to the slope. Let's call this force F_parallel. The force of gravity can be split into two components: the gravitational force pulling the skier downwards (F_down) and the gravitational force perpendicular to the slope (F_perpendicular). The perpendicular component of gravity does not contribute to the skier's motion along the slope. The normal force (N) exerted by the slope cancels out the perpendicular component of gravity.
Therefore, F_parallel = F_down = m * g * sin(θ), where m is the mass of the skier and θ is the angle of the slope (24°).
By Newton's second law, F_parallel = m * a, where a is the acceleration. Thus, we have m * g * sin(θ) = m * a, where g is the acceleration due to gravity (9.8 m/s²).
The mass cancels out, so the acceleration of the skier when going up the hill is a = g * sin(θ).
(b) When the skier has reached the highest point:
At the highest point of the skier's motion, the slope is now perpendicular to the gravitational force. Therefore, there is no force pulling the skier along the slope and the acceleration becomes zero.
(c) After the skier has started sliding down the hill:
When the skier starts sliding down the hill, the direction of the gravitational force component parallel to the slope changes. It now accelerates the skier down the hill. Similar to part (a), the acceleration can be calculated by considering the gravitational force component parallel to the slope.
F_parallel = F_down = m * g * sin(θ)
Since the skier is now moving in the opposite direction, the acceleration is in the opposite direction as well. Hence, the acceleration of the skier sliding down the hill is a = -g * sin(θ).
In summary:
(a) When going up the hill: a = g * sin(θ)
(b) At the highest point: a = 0
(c) After starting sliding down: a = -g * sin(θ)