A skier is accelerating down a 30.0° hill at 1.80 m/s² (Fig. 3–42). How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation change is 125 m?
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검증된 단계별 안내

Step 1: Identify the known values from the problem. The skier's acceleration is \( a = 1.80 \; \text{m/s}^2 \), the angle of the hill is \( \theta = 30.0^\circ \), the elevation change is \( \Delta y = 125 \; \text{m} \), and the skier starts from rest, so \( v_0 = 0 \; \text{m/s} \).

Step 2: Relate the elevation change to the length of the hill. The elevation change \( \Delta y \) is the vertical component of the hill's length \( L \). Using trigonometry, \( \Delta y = L \sin(\theta) \). Rearrange to solve for \( L \): \( L = \frac{\Delta y}{\sin(\theta)} \).

Step 3: Use the kinematic equation to find the time \( t \). The equation \( L = v_0 t + \frac{1}{2} a t^2 \) applies because the skier starts from rest and accelerates uniformly. Substituting \( v_0 = 0 \), this simplifies to \( L = \frac{1}{2} a t^2 \). Rearrange to solve for \( t \): \( t = \sqrt{\frac{2L}{a}} \).

Step 4: Substitute \( L \) from Step 2 into the equation for \( t \). This gives \( t = \sqrt{\frac{2 \cdot \frac{\Delta y}{\sin(\theta)}}{a}} \).

Step 5: Plug in the known values \( \Delta y = 125 \; \text{m} \), \( \theta = 30.0^\circ \), and \( a = 1.80 \; \text{m/s}^2 \) into the equation from Step 4 to calculate \( t \). Ensure that the angle is in degrees or converted to radians if necessary for calculations.

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주요 개념

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

Uniform Acceleration

Uniform acceleration refers to a constant change in velocity over time. In this scenario, the skier accelerates down the hill at a steady rate of 1.80 m/s². This concept is crucial for applying kinematic equations, which relate distance, initial velocity, final velocity, acceleration, and time.

Kinematic Equations

Kinematic equations describe the motion of objects under uniform acceleration. The relevant equation for this problem is d = v_i * t + 0.5 * a * t², where d is the distance traveled, v_i is the initial velocity, a is the acceleration, and t is the time. Since the skier starts from rest, the initial velocity (v_i) is zero, simplifying the equation.

Vertical Distance and Trigonometry

The vertical distance the skier descends is related to the angle of the hill and the length of the slope. Using trigonometric functions, the height (125 m) can be related to the length of the hill. This relationship is essential for determining the total distance the skier travels, which is necessary for calculating the time taken to reach the bottom.

A skier is accelerating down a 30.0° hill at 1.80 m/s² (Fig. 3–42). How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation change is 125 m? <IMAGE>