3.2

Jump to Book Cover

Previous Page

Next Page

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

Solving free fall problems with energy conservation

If you drop a ball, what is its speed just before hitting the ground? One way to answer this question would be to use the equations of motion from Chapter 2. Start with the distance equation and solve for time:

\begin{matrix} h = v_{0} t + \frac{1}{2} g t^{2} = \frac{1}{2} g t^{2} & \Rightarrow & t = \sqrt{\frac{2 h}{g}} \end{matrix}

where v₀ = 0, because it starts from rest. Then use the speed equation and substitute for time:

\begin{matrix} v = v_{0} + g t = g \sqrt{\frac{2 h}{g}} & \Rightarrow & v = \sqrt{2 g h} \end{matrix}

As is often the case in physics, however, the same problem can be solved in another way: using conservation of energy. The ball has gravitational potential energy when it is dropped that is subsequently converted to kinetic energy as it falls.

A ball falls from a height of 8 m. How fast is it traveling when it hits the ground?

4 m/s
13 m/s
78 m/s
157 m/s

Show

From the formula at the top of this page:

\begin{array}{l} v & = & \sqrt{2 g h} \\ = & \sqrt{2 (9.8 N / kg) (8 m)} \\ = & 13 m / s \end{array}

Solved Problem 3.2: Velocity of a falling baseball

A player drops a ball from rest from his outstretched arm. How fast is the ball moving when it has fallen a distance h?

The speed v of the ball just before hitting the ground. (“How fast is it moving?” is another way of asking for speed.)

Determining the speed of a baseball using conservation of energy

Height h the ball has fallen.

Gravitational potential energy: E_p=mgh
Kinetic energy: E_k=½mv²
Conservation of energy: E_start=E_end

To answer the question, let the ball be the system. The force
of gravity is the only influence. The ball can have both
potential energy and kinetic energy but the total energy must
remain constant, E_start=E_end. For simplicity, let the total
energy of the ball be zero before it is dropped, E_start=0. If
the ball falls a height h it changes its potential energy by an
amount −mgh. The total energy of the system, however,
must be zero. Therefore the ball must gain exactly the same
amount of kinetic energy to compensate!

\begin{matrix} E_{e n d} & = & E_{p} + E_{k} & = 0 \\ = & - m g h + \frac{1}{2} m v^{2} & = 0 & \to & \frac{1}{2} m v^{2} = m g h \end{matrix}

With a little algebra we can solve this equation for the speed
of the ball. First, we multiply both sides of the equation
by (2/m) and cancel terms:

\begin{matrix} \frac{2}{m} (\frac{1}{2} m v^{2}) = \frac{2}{m} (m g h) & \to & \frac{2}{m} (\frac{1}{2} m v^{2}) = \frac{2}{m} (m g h) & \to & v^{2} = 2 g h \end{matrix}

Then we take the square root of both sides and arrive at the answer for the speed of the ball v:

\begin{matrix} \sqrt{v^{2}} = \sqrt{2 g h} & \to & v = \sqrt{2 g h} \end{matrix}

The speed of the baseball is v=

\sqrt{2 g h}

after it has fallen by a height h.

Discussion. Note that the answer does not depend on the mass of the baseball! We have used energy conservation to show that objects will all fall at the same rate, independent of their mass.

Extension. It is easy to use this equation to calculate the ball's speed for various heights. If the ball falls 1 meter its speed will be 4.4 m/s. If it falls 2 meters its speed will be 6.3 m/s. After dropping 3 meters the speed is 7.7 m/s. In short, knowing only the height h that it has dropped, we can predict the speed of the ball at any point along its fall.

**Baseball speeds for various dropped heights**
Height h [m]	1	2	3	4	6	8	10
Speed v [m/s]	4.43	6.23	7.67	8.85	10.8	12.5	14.0