Chapter 3: Problem 13

When you throw a ball, the work you do to accelerate it equals the kinetic energy the ball gains. If you do twice as much work when throwing the ball, does it go twice as fast? Explain.

Short Answer

Expert verified

Answer: No, doing twice as much work on a ball does not make it go twice as fast. Instead, the final velocity (v2) is the initial velocity (v1) multiplied by the square root of 2.

Step by step solution

Write down the work-energy theorem

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. Mathematically, it can be written as: W = ΔK.E Where W is the work done, and ΔK.E represents the change in kinetic energy.

Write down the formula for kinetic energy

The kinetic energy of an object can be calculated using the following formula: K.E = (1/2)mv^2 Where K.E denotes the kinetic energy, m is the mass of the object, and v is its velocity.

Set up the equations for the initial and final work done

Let W1 be the initial work done, and let W2 be twice the initial work (W1), i.e., W2 = 2W1. Similarly, let ΔK.E1 be the initial change in kinetic energy and let ΔK.E2 be the final change in kinetic energy after doubling the work done. Using the work-energy theorem, we can write these equations: W1 = ΔK.E1 W2 = ΔK.E2

Calculate the ratio of the initial and final velocities

Since we know that W2 = 2W1, we can write the equation about ΔK.E as follows: 2ΔK.E1 = ΔK.E2 Now, if v1 is the initial velocity and v2 is the final velocity when twice the work is done, we can rewrite the above equation in terms of the kinetic energy formula: 2 * (1/2)m(v1)^2 = (1/2)m(v2)^2 Dividing both sides by m and simplifying, we get: 2(v1)^2 = (v2)^2

Analyze the result

We can take the square root of both sides to find the velocities' relation: √2 * v1 = v2 This result shows that when twice the work is done, the final velocity (v2) is NOT twice the initial velocity (v1). Instead, it is the initial velocity (v1) multiplied by the square root of 2. Thus, the ball does not go twice as fast when twice the work is done.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

When you throw a ball, the work you do to accelerate it equals the kinetic energy the ball gains. If you do twice as much work when throwing the ball, does it go twice as fast? Explain.

Short Answer

Step by step solution

Write down the work-energy theorem

Write down the formula for kinetic energy

Set up the equations for the initial and final work done

Calculate the ratio of the initial and final velocities

Analyze the result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Quantum Physics

Magnetism and Electromagnetic Induction

Electric Charge Field and Potential

Nuclear Physics

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.