Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 19

Chapter 10, Problem 19

As a 15,000 kg jet plane lands on an aircraft carrier, its tail hook snags a cable to slow it down. The cable is attached to a spring with spring constant 60,000 N/m. If the spring stretches 30 m to stop the plane, what was the plane's landing speed?

Verified step by step guidance

Step 1: Recognize that the problem involves energy conservation. The plane's kinetic energy is converted into elastic potential energy stored in the spring as it stretches. Use the principle of energy conservation: $ KE_{initial} = PE_{spring} $.

Step 2: Write the formula for the plane's initial kinetic energy: $ KE_{initial} = \frac{1}{2} m v^2 $, where $ m $ is the mass of the plane (15,000 kg) and $ v $ is its landing speed.

Step 3: Write the formula for the elastic potential energy stored in the spring: $ PE_{spring} = \frac{1}{2} k x^2 $, where $ k $ is the spring constant (60,000 N/m) and $ x $ is the stretch of the spring (30 m).

Step 4: Set $ KE_{initial} $ equal to $ PE_{spring} $: $ \frac{1}{2} m v^2 = \frac{1}{2} k x^2 $. Cancel out the $ \frac{1}{2} $ on both sides to simplify the equation to $ m v^2 = k x^2 $.

Step 5: Solve for $ v $ (the landing speed of the plane) by isolating $ v $: $ v = \sqrt{\frac{k x^2}{m}} $. Substitute the given values for $ k $, $ x $, and $ m $ into the equation to calculate $ v $.

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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 kinetic energy of the jet plane as it lands is converted into elastic potential energy stored in the spring when it stretches. Understanding this relationship allows us to equate the initial kinetic energy of the plane to the potential energy in the spring at maximum stretch.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity of the object. In this case, the jet plane's landing speed is directly related to its kinetic energy, which must be determined to find how fast the plane was moving before it engaged the spring system.

Elastic Potential Energy

Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. It can be calculated using the formula PE = 1/2 kx², where k is the spring constant and x is the displacement from the equilibrium position. In this problem, the spring's stretch provides a means to quantify the energy absorbed from the plane's motion, allowing for the calculation of the initial speed.

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The elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 41 mm while those of nonathletes stretch only 33 mm. The spring constant of the tendon is the same for both groups, 33 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?

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FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?

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In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. Suppose the dam is 80% efficient at converting the water's potential energy to electrical energy. How many kilograms of water must pass through the turbines each second to generate 50 MW of electricity? This is a typical value for a small hydroelectric dam.

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FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0 m. What is the particle's maximum speed? At what position does it have this speed?

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In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. What is $Δ U_{G}$ of 1.0 kg of water?

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A stretched spring stores 2.0 J of energy. How much energy will be stored if the spring is stretched three times as far?

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