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 17

Chapter 8, Problem 17

A 1400-kg car moving on a horizontal surface has speed v = 85 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring constant of the spring? Ignore any thermal energy produced in the collision.

Verified step by step guidance

Convert the car's speed from km/h to m/s. Use the conversion factor: 1 km/h = 1000 m / 3600 s. Thus, $ v = 85 \; \text{km/h} $ can be converted to $ v = \frac{85 \times 1000}{3600} \; \text{m/s} $.

Apply the work-energy principle. The car's initial kinetic energy is completely converted into the elastic potential energy of the spring. The equation is $ \frac{1}{2} m v^2 = \frac{1}{2} k x^2 $, where $ m $ is the mass of the car, $ v $ is its initial velocity, $ k $ is the spring constant, and $ x $ is the compression distance of the spring.

Simplify the equation to solve for the spring constant $ k $: $ k = \frac{m v^2}{x^2} $.

Substitute the known values into the equation: $ m = 1400 \; \text{kg} $, $ v $ (in m/s from step 1), and $ x = 2.2 \; \text{m} $.

Perform the calculations to find the value of $ k $, ensuring all units are consistent (mass in kg, velocity in m/s, and distance in m).

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Key Concepts

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

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 the mass and v is the velocity. In this scenario, the car's kinetic energy before it strikes the spring will be converted into potential energy stored in the spring as it compresses.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This principle is essential for determining the spring constant when the spring is compressed by the car.

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 problem, the kinetic energy of the car is transformed into elastic potential energy in the spring, allowing us to equate the two to find the spring constant.

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