Ch. 03 - Kinematics in Two or Three Dimensions; Vectors

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 03 - Kinematics in Two or Three Dimensions; Vectors

Problem 23b

Chapter 3, Problem 23b

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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Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Related Practice

Textbook Question

A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp (Fig. 3–46). With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 m above the car roofs and the horizontal distance he must clear is 22 m.

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Textbook Question

At serve, a tennis player aims to hit the ball horizontally. What minimum speed is required for the ball to clear the 0.90-m-high net about 15.0 m from the server if the ball is 'launched' from a height of 2.30 m? Where will the ball land if it just clears the net (and will it be 'good' in the sense that it lands within 7.0 m of the net)? How long will it be in the air? See Fig. 3–50.

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Textbook Question

A car is moving with speed 16.0 m/s due south at one moment and 25.7 m/s due east 8.00 s later. Over this time interval, determine the magnitude and direction of (a) its average velocity, (b) its average acceleration. (c) What is its average speed? [Hint: Can you determine all these from the information given?]

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Textbook Question

A skier is accelerating down a 30.0° hill at 1.80 m/s² (Fig. 3–42). What is the vertical component of her acceleration?

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Textbook Question

Two vectors, $V ⃗ sub 1$ and $V ⃗ sub 2$ , add to a resultant ${\vec{V}}_{R} = {\vec{V}}_{1} + {\vec{V}}_{2}$ . Describe $V ⃗ sub 1$ and $V ⃗ sub 2$ if $V_{R}^{2} = V_{1}^{2} + V_{2}^{2}$ .

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Textbook Question

A diver running 2.5 m/s dives out horizontally from the edge of a vertical cliff and 3.5 s later reaches the water below. How high was the cliff and how far from its base did the diver hit the water?

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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>

Verified video answer for a similar problem:

Key Concepts

Uniform Acceleration

Kinematic Equations

Vertical Distance and Trigonometry

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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