A skier is gliding along at 3.0 m/s on horizontal, frictionless snow. He suddenly starts down a 10° incline. His speed at the bottom is 15 m/s. What is the length of the incline?

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Identify the known quantities: initial velocity \( v_i = 3.0 \; \text{m/s} \), final velocity \( v_f = 15.0 \; \text{m/s} \), and the angle of the incline \( \theta = 10^\circ \). The goal is to find the length of the incline \( d \).

Use the work-energy principle, which states that the change in kinetic energy is equal to the work done by gravity. The equation is \( \Delta KE = W_g \), where \( \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \) and \( W_g = m g d \sin(\theta) \).

Cancel out the mass \( m \) from both sides of the equation since it appears in every term. This simplifies to \( \frac{1}{2} v_f^2 - \frac{1}{2} v_i^2 = g d \sin(\theta) \).

Rearrange the equation to solve for \( d \): \( d = \frac{v_f^2 - v_i^2}{2 g \sin(\theta)} \).

Substitute the known values into the equation: \( v_f = 15.0 \; \text{m/s} \), \( v_i = 3.0 \; \text{m/s} \), \( g = 9.8 \; \text{m/s}^2 \), and \( \sin(10^\circ) \). Perform the calculations to find \( d \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Conservation of Energy

The principle of conservation of energy states that in a closed system, the total energy remains constant. In this scenario, the skier's initial kinetic energy and potential energy at the top of the incline will convert into kinetic energy at the bottom. This concept is crucial for determining the relationship between the skier's speeds and the height of the incline.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this problem, the skier's speed changes from 3.0 m/s to 15 m/s, indicating a change in kinetic energy as he descends the incline, which is essential for calculating the length of the incline.

Inclined Plane Dynamics

Inclined plane dynamics involves analyzing the motion of objects on slopes, taking into account gravitational forces and angles. The angle of the incline affects the component of gravitational force acting along the slope, which influences the skier's acceleration. Understanding this concept helps in determining the length of the incline based on the skier's change in speed.