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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
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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

Elastic collisions

Comparison between elastic, inelastic, and perfectly inelastic collisions

In all collisions, momentum is conserved. In a special type of collision called an elastic collision, however, kinetic energy is also conserved. An elastic collision occurs when a rubber ball bounces off the floor and reaches the same height it was dropped from. Nearly elastic collisions often occur in billiards when a fast-moving cue ball strikes a numbered ball, causing the cue ball to stop in place and the numbered ball to move off in the same direction.

\frac{1}{2} m_{1} v_{_{i 1}}^{2} + \frac{1}{2} m_{2} v_{_{i 2}}^{2} = \frac{1}{2} m_{1} v_{_{f 1}}^{2} + \frac{1}{2} m_{2} v_{_{f 2}}^{2}

Conservation of energy
for elastic collisions

In inelastic collisions, only momentum is conserved. With one equation describing the motion of the colliding objects, a problem can only be solved uniquely when there is one unknown variable. One equation, one unknown variable. In an elastic collision, however, there are two equations of motion: conservation of momentum and conservation of kinetic energy. An elastic collision problem can therefore be more difficult, because it may involve two equations and two unknown variables. Because of the complexity of elastic collisions, problems you will encounter will usually have only one unknown quantity—which means that you can use momentum conservation alone. A problem may require using the energy conservation equation, however, to determine if a given collision is elastic or inelastic. Show Quadratic solution to elastic collision problems

Show Quadratic solution to elastic collision problems

What can make an elastic collision problem even more difficult is that the conservation of kinetic energy equation is quadratic in speed, i.e., the speed variables are all squared. In algebra class you learned that if you have a quadratic equation of the form

y = a x^{2} + b x + c

then the two solutions (or roots) of the equation are:

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

More difficult elastic collision problems involving these quadratic equations will be covered in more depth in later chapters in the book. Hide me!

Hide me!

In elastic collisions, which of the following quantities are conserved?

Velocity
Momentum
Kinetic energy

I only
I and III only
II and III only
I, II, and III

Show

Solved Problem 3.7: Is this collision elastic?

A 0.16 kg cue ball traveling at 4 m/s strikes a stationary 0.16 kg eight-ball. After the collision, the cue ball travels at 0.2 m/s while the eight-ball travels at 3.8 m/s. Is this an elastic collision? Why or why not?

Is this collision elastic?

If it is an elastic collision. In other words, is kinetic energy conserved in the collision?

Mass of the cue ball and eight-ball m₁ = m₂ = 0.16 kg;
initial speed of the cue ball v_i,1 = 4 m/s; initial speed of the eight-ball v_i,2 = 0 m/s; final speed of the cue ball v_f,1 = 0.2 m/s; and final speed of the eight-ball v_f,1 = 3.8 m/s.

Conservation of energy for two objects in an elastic collision:

\frac{1}{2} m_{1} v_{_{i 1}}^{2} + \frac{1}{2} m_{2} v_{_{i 2}}^{2} = \frac{1}{2} m_{1} v_{_{f 1}}^{2} + \frac{1}{2} m_{2} v_{_{f 2}}^{2}

Calculate the kinetic energy of each ball both before and after the collision and see if they are equal:

\begin{matrix} \frac{1}{2} (0.16 kg) {(4 m/s)}^{2} + \frac{1}{2} (0.16 kg) {(0)}^{2} & \overset{?}{=} & \frac{1}{2} (0.16 kg) {(0.2 m/s)}^{2} + \frac{1}{2} (0.16 kg) {(3.8 m/s)}^{2} \\ 1.28 J + 0 J & \overset{?}{=} & 0 .003 J + 1.155 J \end{matrix}

Some kinetic energy is lost by the collision, therefore the collision is not elastic.

The collision is not elastic because kinetic energy is lost during the collision. (Since only a small amount of the kinetic energy is lost, the collision could be called “nearly elastic.”)