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Chapter 5: Problem 27

Apply How does the kinetic energy of an object change if its speed doubles? Triples?

Short Answer

Expert verified
If speed doubles, kinetic energy becomes four times greater; if speed triples, kinetic energy becomes nine times greater.

Step by step solution

01

Understanding Kinetic Energy

The kinetic energy of an object is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. Kinetic energy is directly proportional to the square of the velocity.
02

Doubling Speed Analysis

If the speed \( v \) of the object doubles, it becomes \( 2v \). Substitute \( 2v \) into the kinetic energy formula: \( KE = \frac{1}{2}m(2v)^2 = \frac{1}{2}m(4v^2) = 2mv^2 \). The kinetic energy becomes four times the original.
03

Tripling Speed Analysis

If the speed \( v \) of the object triples, it becomes \( 3v \). Substitute \( 3v \) into the kinetic energy formula: \( KE = \frac{1}{2}m(3v)^2 = \frac{1}{2}m(9v^2) = 4.5mv^2 \). The kinetic energy becomes nine times the original.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity
Velocity refers to how fast an object is moving in a certain direction. It is a vector quantity, which means that it has both a magnitude (how fast) and a direction (which way). For example, if a car is traveling at 60 km/h towards the north, its velocity is described by both the speed (60 km/h) and the direction (north).
  • Velocity is different from speed, as speed doesn鈥檛 include direction.
  • Changing either the speed or the direction changes the velocity.
  • Velocity is crucial when calculating kinetic energy, as it affects how much energy an object in motion can have.
Kinetic Energy Formula
The kinetic energy of an object in motion can be calculated using the formula: \[ KE = \frac{1}{2}mv^2 \] Here, \( KE \) stands for kinetic energy, \( m \) is the mass of the object, and \( v \) is its velocity. What this formula tells us is that kinetic energy depends on both the mass and the velocity but is more significantly affected by changes in velocity due to its square. For example, if an object's velocity doubles, its kinetic energy will increase by four times, highlighting just how influential velocity is in determining kinetic energy.
  • The formula explains why faster-moving objects have much more kinetic energy.
  • A small increase in velocity means a larger increase in kinetic energy.
Proportionality
Proportionality in the context of kinetic energy means that some relationships in the formula change according to consistent ratios. For kinetic energy, specifically, the relationship between kinetic energy and velocity is one of square proportionality. This means:
  • If the velocity of an object doubles (\( v \to 2v \)), then the kinetic energy becomes four times greater because: \[ KE = \frac{1}{2}m(2v)^2 = 2mv^2 \]
  • If the velocity triples (\( v \to 3v \)), then the kinetic energy increases by nine times the original: \[ KE = \frac{1}{2}m(3v)^2 = 4.5mv^2 \]
Understanding this proportionality helps predict how the energy of an object will change with velocity changes, making it a key concept in physical science.
Speed
Speed refers to how fast an object is moving, regardless of the direction. It is a scalar quantity, which means it only has a magnitude but no specific direction. Often, speed is the first step toward understanding more complex concepts like velocity and kinetic energy. For example:
  • A car traveling at 70 km/h has a speed of 70 km/h.
  • Unlike velocity, speed does not tell us where the object is heading.
  • Speed plays an important role in everyday scenarios and serves as a building block to understand kinetic energy calculations.
When we talk about kinetic energy, the changes in speed become crucial because they significantly affect the energy a moving object has. Speed is typically what people refer to when discussing how fast something is going, making it a simpler but essential concept.

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Most popular questions from this chapter

A record for running up the stairs of the Empire State Building was set on February 3, 2003. The runner completed the 86 flights, with a total of 1576 steps, in 9 min \(33 \mathrm{~s}\). If the altitude gain for each step was \(0.20 \mathrm{~m}\) and the mass of the runner was \(70.0 \mathrm{~kg}\), what was his average power output during the climb? Give your answer in both watts and horsepower.

Assess System 1 has a force of \(10 \mathrm{~N}\) and a speed of \(5 \mathrm{~m} / \mathrm{s}\). System 2 has a force of \(20 \mathrm{~N}\) and a speed of \(2 \mathrm{~m} / \mathrm{s}\). Which system has the greater power? Explain.

Calculate A 9.50-g bullet has a speed of \(1.30 \mathrm{~km} / \mathrm{s}\). What is the kinetic energy of the bullet?

A package rests on the floor of an elevator that is rising with constant speed. The elevator exerts an upward force on the package and thus does positive work on it. Why doesn't the kinetic energy of the package increase?

Is it possible for the kinetic energy of an object to be negative? Explain.
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