Question

A skier starts down a 12° incline at 2.0 m/s, reaching a speed of 17 m/s at the bottom. Friction between the snow and her freshly waxed skis is negligible. What is the length of the incline? How long does it take the skier to reach the bottom?

Answer 1

1. The length of the incline is approximately 109 meters.

2. The skier takes about 22 seconds to reach the bottom.

Step-by-step explanation:

The key to solving this problem lies in applying the principles of kinematics and using the equations of motion.

1. Length of the Incline: We can use the kinematic equation
`\(v_f^2 = v_i^2 + 2a d\)`, where
`\(v_f\)` is the final velocity,
`\(v_i\)` is the initial velocity, a is the acceleration, and d is the displacement. Since the skier is moving downhill, the acceleration is due to gravity and is given by
`\(a = g \sin(\theta)\)`, where g is the acceleration due to gravity (approximately 9.8 m/s²) and
`\(\theta\)` is the angle of the incline.

Substituting the known values, we can solve for d.

`\[ d = \frac{{v_f^2 - v_i^2}}{{2a}} \]`

`\[ d = \frac{{(17 \, \text{m/s})^2 - (2.0 \, \text{m/s})^2}}{{2 * 9.8 \, \text{m/s}^2 * \sin(12^\circ)}} \]`

`\[ d \approx 109 \, \text{meters} \]`

2. Time to Reach the Bottom: We can use the kinematic equation
`\(d = v_i t + (1)/(2) a t^2\)` to find the time (t) it takes for the skier to reach the bottom.

Rearranging the equation and solving for t, we get:

`\[ t = \frac{{v_f - v_i}}{{a}} \]`

`\[ t = \frac{{17 \, \text{m/s} - 2.0 \, \text{m/s}}}{{9.8 \, \text{m/s}^2 \sin(12^\circ)}} \]`

`\[ t \approx 22 \, \text{seconds} \]`

In summary, the length of the incline is approximately 109 meters, and it takes the skier about 22 seconds to reach the bottom.

Full Question:

A skier starts down a 12 ∘ incline at 2.0 m/s, reaching a speed of 17 m/s at the bottom. Friction between the snow and her freshly waxed skis is negligible.

1. What is the length of the incline?

2. How long does it take the skier to reach the bottom?