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Videos: Conservation of energy Elastic potential energy Elastic potential energy in your bicep and tricep muscles Energy and work Gravitational potential energy Kinetic energy Physics of wind power Solar flare is a plasma in motion Steam rising out of cooling tower at a nuclear power plant

Investigations: 3A: Work and the force vs. distance graph (p. 113) 3B: Inclined plane and the conservation of energy (p. 118) 3C: Work and energy for launching a paper airplane (p. 122) 3D: Springs and the conservation of energy (p. 124) 3E: Conservation of momentum (p. 128) 3F: Collisions (p. 132) 3G: Shock absorbers and dampers (p. 139) 3H: Specific heat of water and steel (p. 144)

Science: Algebra derivation using the conservation of energy Brownian motion Choosing origin for potential energy reference frame Difference between heat and temperature Einstein's theory of relativity Elastic potential energy Elastic potential energy in your muscles Elastic potential energy in your muscles Energy and power for light bulb Energy and work Energy content of phases of matter Energy flow in open and closed systems Faster than light travel? Forms of energy Friction Galileo and the Leaning Tower of Pisa Gravitational potential energy Hooke's Law for springs Intensity of sunlight at Earth's surface Kelvin temperature scale and absolute zero Kinetic energy Law of conservation of energy Loose springs and stiff springs (BLM) Ordered and random kinetic energy Other conservation laws in physics Plasma in motion during a solar flare Plasma is a phase of matter Potential energy Power and radiant energy Power is the rate of doing work Radiant energy Relationship between momentum conservation and Newton's third law Symmetry in the physics of collisions Units of work and energy van der Waals forces

Technology: Batteries and electrical power Calories on food labels Conserving energy in everyday life Converting units and understanding energy content from food Efficiency in technology Energy content of everyday goods Energy flow in clothing manufacture Energy transformations in an internal combustion (piston) engine Energy transformations in technology usually produce heat Everyday objects that store energy Everyday technology in our age of energy Friction from brake pads in a car's wheel Hydroelectric dam Objects with elastic potential energy Power in everyday technology Power, wattage, and the light bulb Radiant power and the light bulb Technology and agriculture over the last century

Engineering: Calculating energy production for the Hoover Dam Design a wind turbine power plant Efficiency Electric current Electromagnetic spectrum Open and closed systems Siting constraint for power plants Specific heat Springs Volts, amps, and power

Mathematics: Algebra derivation using the conservation of energy Calculating energy production for the Hoover Dam Changes in a quantity represented by Greek symbol Δ Choosing origin for potential energy reference frame Converting units and understanding energy content from food Deducing a physical equation Different rates of change Hooke's Law for springs Non-linear relationships Proportional relationships Using algebra to derive temperature conversion formula Using algebra to solve for a variable

Interactive Simulations Interactive simulation of motion down an inclined plane Motion down an inclined plane Interactive simulation of an oscillating spring Spring-loaded carts simulation Elastic collisions between two balls Inelastic collisions between two balls Interactive simulation of an oscillating spring Interactive simulation of an oscillating spring Wind power design simulation Interactive Calculators Work equation calculator Kinetic energy equation Potential energy equation Elastic potential energy equation Efficiency equation Conservation of momentum in collisions equation Power equation calculator
Electric power equation Temperature conversion equation Specific heat equation

1 - Science of Physics
2 - Forces and Motion 3 - Energy Transformations Chapter 3 at a glance 3.1 - Energy 3.2 - Conservation of energy 3.3 - Flow of energy 3.4 - Temperature and the kinetic theory of matter 3.5 - Chapter review4 - Waves and Sound
5 - Electricity and Magnetism
6 - Light and Optics
7 - The Atom
8 - Linear Motion
9 - Vectors and Forces
10 - Forces
11 - Newton's Laws and Gravitation
12 - Circular Motion and Gravitation
13 - Torque and Static Equilibrium
14 - Momentum and Collisions
15 - Angular momentum
16 - Power and Machines
17 - Harmonic Motion
18 - Waves
19 - Sound
20 - Electricity
21 - Electric and Magnetic Fields
22 - Electromagnetism and Induction
23 - Light and Color
24 - Geometrical Optics
25 - Electromagnetic Radiation
26 - The Properties of Matter
27 - Thermodynamics and Heat Transfer
28 - The Atom
29 - The Quantum Theory
30 - TBD
31 - TBD
32 - TBD
33 - TBD
34 - TBD
35 - TBD
36 - TBD
37 - The atomic nucleus and radioactivity
38 - Nuclear reactions
39 - Relativity
40 - Particle physics and the standard model
41 - Electronics (semiconductors)
42 - Nuclear technology
43 - Lasers and photonics
44 - Metals, alloys and materials science
45 - Communications technology
46 - Buildings and structures
47 - Energy technology
48 - Biophysics
49 - Matter and Energy, Space and Time
50 - Energy Transformations
51 - Nanotechnology
52 - Website Videos
53 - ExtraStuff
55 - Modeling Motion, Friction, and Center of Mass
56 - Work and the Conservation of Energy

Kinetic energy

Kinetic energy (KE) is energy of motion. Any object that has mass and is moving has kinetic energy because it is moving. When you catch a ball, your hand applies a force over some distance (your hand recoils a bit) to stop the ball. That force multiplied by the distance represents the transfer of the ball’s kinetic energy to your hand by doing work on your hand. Now think about catching balls of different masses and speeds. You can probably guess that the kinetic energy of a massive or a fast-moving ball has more kinetic energy—it is harder to stop!—than a lighter, slower ball.

E_{k} = \frac{1}{2} m v^{2}

E_k	=	kinetic energy (J)
m	=	mass (kg)
v	=	speed (m/s)

The kinetic energy of a moving object is proportional to mass. If you double the mass, you double the kinetic energy. For example, a 2 kg ball moving at a speed of 1 m/s has 1 joule of kinetic energy according to equation (3.2). A 4 kg ball moving at the same speed has 2 joules of kinetic energy, or twice as much. This is a linear relationship—when mass doubles the kinetic energy also doubles.

How kinetic energy is proportional to mass and proportional to speed squared

According to equation (3.2) kinetic energy depends on the square of the speed of a moving object. Consider a 2 kilogram ball traveling at 1 m/s, with 1 joule of kinetic energy. The same ball moving at 3 m/s has nine joules of kinetic energy. If you multiply the speed by three, then the kinetic energy is multiplied by a factor of 3² = 9. This is called a non-linear relationship. Show Non-linear relationships

Show Non-linear relationships

$Linear and non-linear relationships in math and physics$ A linear relationship—i.e., a straight line on an x–y graph—can be represented by the formula y = mx + b . If you change x by a factor of two, then y changes by a factor of 2m; if you change x by a factor of three, then y changes by a factor of 3m; and so on. In this linear relationship, y is a function of x—where x is raised to the first power.

In a non-linear relationship, however, y is a function of x raised to some other power—such as x², or x⁷, or

\sqrt{x}

. While the x–y graph of linear relationship such as y = x is a straight line, the x–y graph of non-linear relationship such as y = x² is a curved line (a parabola in this case). The equation for kinetic energy:

shows that kinetic energy is a non-linear function of velocity v (because it depends on velocity squared v²), but it is a linear function of mass m. Hide me!

Hide me!

The fact that kinetic energy increases with the square of the speed has implications for the stopping distance of a car. When it stops, the car's kinetic energy is transformed into thermal energy—heat—through work done by the brakes. Work is force multiplied by distance. Car brakes provide a relatively constant force, therefore the distance it takes the car to stop is proportional to the kinetic energy. At 30 mph a car can stop in 16 meters—around 4 body lengths. At twice the speed, or 60 mph, it takes four times longer to stop.

Braking distance increases as the square of the car's speed

A motorcycle has 10 joules of kinetic energy. If it's velocity is doubled, how much kinetic energy does it now possess?

15 J
20 J
35 J
40 J

Show