Ch. 08 - Conservation of Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 08 - Conservation of Energy

Problem 11

Chapter 8, Problem 11

A novice skier, starting from rest, slides down an icy frictionless 8.0° incline whose vertical height is 115 m. How fast is she going when she reaches the bottom?

Verified step by step guidance

Step 1: Identify the given values and the goal of the problem. The skier starts from rest, so the initial velocity $ v_0 $ is 0. The incline angle is $ \theta = 8.0^\circ $, and the vertical height $ h $ is 115 m. The goal is to find the final velocity $ v_f $ at the bottom of the incline.

Step 2: Use the principle of conservation of energy. Since the incline is frictionless, the skier's potential energy at the top is completely converted into kinetic energy at the bottom. The formula for conservation of energy is $ m g h = \frac{1}{2} m v_f^2 $, where $ m $ is the mass of the skier, $ g $ is the acceleration due to gravity ($ 9.8 \ \text{m/s}^2 $), $ h $ is the vertical height, and $ v_f $ is the final velocity.

Step 3: Simplify the equation by canceling out the mass $ m $, since it appears on both sides of the equation. This gives $ g h = \frac{1}{2} v_f^2 $. Rearrange the equation to solve for $ v_f $: $ v_f = \sqrt{2 g h} $.

Step 4: Substitute the known values into the equation. Use $ g = 9.8 \ \text{m/s}^2 $ and $ h = 115 \ \text{m} $. The equation becomes $ v_f = \sqrt{2 \cdot 9.8 \cdot 115} $.

Step 5: Perform the calculation to find $ v_f $. This will give the skier's final velocity at the bottom of the incline. Ensure the units are consistent throughout the calculation.

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Key Concepts

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

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the skier's potential energy at the top of the incline is converted into kinetic energy as she slides down. This relationship allows us to calculate the skier's speed at the bottom by equating the initial potential energy to the final kinetic energy.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. For the skier, the potential energy at the top of the incline can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. This energy is what drives the skier's motion down the slope.

Kinetic Energy

Kinetic energy is the energy of an object in motion, defined by the formula KE = 0.5mv², where m is mass and v is velocity. As the skier descends the incline, her potential energy is converted into kinetic energy, which determines her speed at the bottom of the slope. Understanding this relationship is crucial for solving the problem.

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