3.2

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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

Inelastic collisions

Imagine a collision between two objects with masses m₁ and m₂. The two objects have initial velocities v_i1 and v_i2 and final velocities v_f1 and v_f2. Momentum conservation for these two colliding objects can be written as:

m_{1} v_{i 1} + m_{2} v_{i 2} = m_{1} v_{f 1} + m_{2} v_{f 2}

m₁	=	Mass, object 1 (kg)
v_i,1	=	Initial speed, object 1 (m/s)
v_f,1	=	Final speed, object 1 (m/s)
m₂	=	Mass of object 1 (kg)
v_i,2	=	Initial speed, object 2 (m/s)
v_f,2	=	Final speed, object 2 (m/s)

Conservation
of momentum
(two objects)

There are two types of collisions in physics: elastic and inelastic. In an inelastic collision, some of the initial kinetic energy of the objects is transformed into heat and/or deforming the shape of the objects. Auto collisions are nearly always inelastic, because of the damage caused to the cars. In the special case of a perfectly inelastic collision, the two objects stick together after impact. Show Symmetry and collisions

Show Symmetry and collisions

Perfectly inelastic collision between two balls

A perfectly inelastic collision is depicted in the illustration above. These collision problems are solved in the same way as any other collision problem, using the conservation of momentum. In the perfectly inelastic collision case, however, the final velocities of the two objects are set to be equal—because they stick together!

Two cars collide head-on, partially crumpling the front end of each. The cars bounce off each other from the collision, ending 1.3 m apart. What kind of collision is this?

Elastic
Inelastic
Perfectly inelastic
Not enough information

Show

Two 1 kg balls, traveling at +1 m/s and −1 m/s, collide with each other and stick together after impact. What is their speed after the collision?

−1 m/s
0 m/s
+1 m/s
+2 m/s

Show

Solved Problem 3.6: Final speed in a perfectly inelastic collision

A 100 kg hockey player, moving at 2 m/s collides, with a 75 kg hockey player, moving at 1 m/s. Both players were moving towards each other. After impact, they become entangled and slide together. What is their speed after impact?

Final speed after collision v_f

Mass of first player m₁ = 100 kg; Mass of second player m₁ = 75 kg;
Initial speed of first player v_i,1 = +2 m/s; (choose his direction as the positive direction)
Initial speed of first player v_i,1 = -1 m/s (negative because he is moving towards the first player).

Conservation of momentum:

m_{1} v_{i 1} + m_{2} v_{i 2} = m_{1} v_{f 1} + m_{2} v_{f 2}

Since they move together after the (inelastic) collision, the final speeds are the same v_f,1 = v_f,2 = v_f:

m_{1} v_{i, 1} + m_{2} v_{i, 2} = m_{1} v_{f, 1} + m_{2} v_{f, 2} = m_{1} v_{f} + m_{2} v_{f} = (m_{1} + m_{2}) v_{f}

Divide both sides by (m₁ + m₂):

\frac{m_{1} v_{i, 1} + m_{2} v_{i, 2}}{m_{1} + m_{2}} = \frac{(m_{1} + m_{2}) v_{f}}{m_{1} + m_{2}}

Cancel terms to solve for v_f and substitute the values:

v_{f} = \frac{m_{1} v_{i, 1} + m_{2} v_{i, 2}}{m_{1} + m_{2}} = \frac{(100 kg) (2 m/s) + (75 kg) (- 1 m/s)}{100 kg + 75 kg} = + 0.71 m/s

v_f = +0.71 m/s. (Positive value means in the same direction as the 100 kg player.)