A slingshot will shoot a 1010-g pebble 22.022.0 m straight up. How much potential energy is stored in the slingshot's rubber band?

Ch 07: Potential Energy & Conservation

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 07: Potential Energy & Conservation

Problem 18a

Chapter 7, Problem 18a

A slingshot will shoot a $10$ -g pebble $22.0$ m straight up. How much potential energy is stored in the slingshot's rubber band?

Verified step by step guidance

Step 1: Identify the key variables in the problem. The mass of the pebble is 10 g (convert to kilograms: 0.010 kg), the height it reaches is 22.0 m, and we are solving for the potential energy stored in the slingshot's rubber band.

Step 2: Recall the formula for gravitational potential energy, which is given by $ U = m g h $, where $ m $ is the mass, $ g $ is the acceleration due to gravity (approximately $ 9.8 \ \text{m/s}^2 $), and $ h $ is the height.

Step 3: Substitute the known values into the formula. Use $ m = 0.010 \ \text{kg} $, $ g = 9.8 \ \text{m/s}^2 $, and $ h = 22.0 \ \text{m} $. The equation becomes $ U = (0.010)(9.8)(22.0) $.

Step 4: Understand that the potential energy stored in the slingshot's rubber band is equal to the gravitational potential energy of the pebble at its maximum height, assuming no energy is lost to air resistance or other factors.

Step 5: Perform the multiplication to calculate $ U $, but do not provide the numerical result here. The final value of $ U $ will represent the potential energy stored in the slingshot's rubber band.

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Key Concepts

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

Gravitational Potential Energy

Gravitational potential energy (PE) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and h is the height above a reference point. In this case, the height is the maximum height the pebble reaches, which is 22.0 m.

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 the context of the slingshot, the elastic potential energy stored in the rubber band is converted into gravitational potential energy as the pebble is launched upward. This principle helps in understanding how the energy stored in the slingshot translates into the energy of the pebble at its peak height.

Elastic Potential Energy

Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. For a slingshot, this energy is stored in the rubber band when it is pulled back. The amount of elastic potential energy can be calculated using the formula PE_elastic = 1/2 kx², where k is the spring constant and x is the displacement from the equilibrium position. This energy is what propels the pebble when the slingshot is released.

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Textbook Question

A slingshot will shoot a $10$ -g pebble $22.0$ m straight up. With the same potential energy stored in the rubber band, how high can the slingshot shoot a $25$ -g pebble? What physical effects did you ignore in solving this problem?

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Textbook Question

A spring of negligible mass has force constant $k = 1600$ N/m. How far must the spring be compressed for $3.20$ J of potential energy to be stored in it?

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Textbook Question

A spring of negligible mass has force constant $k equals 800$ N/m. How far must the spring be compressed for $1.20$ J of potential energy to be stored in it?

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Textbook Question

In one day, a $75$ -kg mountain climber ascends from the $1500$ -m level on a vertical cliff to the top at $2400$ m. The next day, she descends from the top to the base of the cliff, which is at an elevation of $1350$ m. What is her change in gravitational potential energy on the first day?

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Textbook Question

Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length $20$ m that makes an angle of $45 degrees$ with the vertical, steps off his tree limb, and swings down and then up to Jane's open arms. When he arrives, his vine makes an angle of $30 degrees$ with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan's speed just before he reaches Jane. Ignore air resistance and the mass of the vine.

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Textbook Question

The maximum height a typical human can jump from a crouched start is about $60$ cm. By how much does the gravitational potential energy increase for a $72$ -kg person in such a jump? Where does this energy come from?

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