Ch. 14 - Oscillations

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. 14 - Oscillations

Problem 75

Chapter 14, Problem 75

An energy-absorbing car bumper has a spring constant of 410 kN/m. Find the maximum compression of the bumper if the car, with mass 1300 kg, collides with a wall at a speed of 2.0 m/s (approximately 5 mi/h).

Verified step by step guidance

Step 1: Identify the type of energy involved in the problem. The car has kinetic energy due to its motion, and this energy will be converted into elastic potential energy stored in the spring of the bumper during compression.

Step 2: Write the formula for kinetic energy: $ KE = \frac{1}{2} m v^2 $, where $ m $ is the mass of the car (1300 kg) and $ v $ is its velocity (2.0 m/s). Substitute the given values into this formula to calculate the car's initial kinetic energy.

Step 3: Write the formula for elastic potential energy stored in a spring: $ PE_{spring} = \frac{1}{2} k x^2 $, where $ k $ is the spring constant (410 kN/m, or 410,000 N/m) and $ x $ is the maximum compression of the spring. This is the energy the spring absorbs during the collision.

Step 4: Apply the principle of energy conservation. Set the car's initial kinetic energy equal to the spring's elastic potential energy: $ \frac{1}{2} m v^2 = \frac{1}{2} k x^2 $. Cancel out the $ \frac{1}{2} $ on both sides of the equation.

Step 5: Solve for $ x $, the maximum compression of the spring. Rearrange the equation to $ x = \sqrt{\frac{m v^2}{k}} $. Substitute the values for $ m $, $ v $, and $ k $ into this formula to find $ x $.

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Key Concepts

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

Spring Constant

The spring constant, denoted as 'k', measures the stiffness of a spring. It is defined as the force required to compress or extend the spring by a unit distance. In this case, a spring constant of 410 kN/m indicates that a force of 410,000 Newtons is needed to compress the bumper by one meter.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v^2, where 'm' is mass and 'v' is velocity. For the car in the question, its kinetic energy at a speed of 2.0 m/s can be determined, which will be converted into potential energy when the bumper compresses.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the kinetic energy of the car will be converted into elastic potential energy stored in the compressed bumper spring, allowing us to find the maximum compression using the relationship between these energy forms.

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