
Einstein's Big Idea


Classroom Activities

Squaring Off With Velocity

Activity Summary
Students investigate the meaning of c^{2} in E =
mc^{2} by measuring the energy delivered by an object falling at
different velocities. Graphing data leads students to understand that E
is proportional to velocity squared, not simply velocity.
Learning Objectives
Students will be able to:
explain what the c^{2} in E = mc^{2}
represents.
state that kinetic energy is the energy of an object in motion.
illustrate how kinetic energy can be transferred to other objects.
understand that the energy delivered by an object in motion is
proportional to v^{2}, not v.
 copy of the "Squaring Off With Velocity" student handout
(PDF or
HTML)
 copy of the "Data Sheet" student handout
(PDF or
HTML)
 1 lb flour in a plastic bag
 plastic pan (about shoebox size)
 two 8oz plastic cups
 four standardsized (about 1 cm) glass marbles
 plastic ruler
 meter stick
 plastic spoon
 wood dowel or wood barbecue skewer
 felttip pen
 graph paper
 newspaper
Background
Light
has fascinated scientists for centuries. Galileo Galilei was the first to
consider measuring its speed. In 1676, astronomer Ole Roemer made observations
of the eclipses of Jupiter's moons to demonstrate that light moved at a very
fast—but not infinite—speed. James Clerk Maxwell provided the
mathematical backbone for electromagnetism and demonstrated that light was an
electromagnetic wave. The squared part of Albert Einstein's
equation heralds back to natural philosopher Gottfried Leibniz, who proposed
that an object's energy is the product of its mass times its velocity squared,
not just its velocity. Emilie du Châtelet further championed his ideas.
While many of these scientists were innovative thinkers, determination also
played a large part in their achievements. They were willing to challenge
widely held beliefs of their day. Their courage and perseverance helped lay the
groundwork for Einstein's eventual connection of mass, energy, and the speed of
light squared.
In this activity, students use a simple model to investigate the relationship
between velocity and energy. Their investigation leads them to conclude that
the energy delivered to a system depends on the velocity squared of
the impacting body, not simply on the velocity. Students then relate this fact
to E = mc^{2}.
The model uses a glass marble as a falling object that impacts a cup of flour.
The impact velocity of the marble is a function of the height from which the
marble is dropped. The energy released by the falling marble (its potential
energy now turned into kinetic energy) is equal to the work done on the flour.
The work done on the flour, in turn, is equal to the force (mass x
deceleration) it takes to slow the marble down to zero velocity over the
distance it penetrates the flour.
Four heights are used10, 25, 50, and 100 centimeters. Students graph the
velocity the marble attains when dropped from these heights against the depth
to which the marble penetrates the flour. The depth, in turn, is a measure of
the energy that the marble delivers. Students can calculate the impact velocity
at these four heights by using the equation:
where g = the acceleration due to
gravity (9.8 m/sec^{2}), and d = the distance the marble falls (in meters)
Key Terms
kinetic
energy: Energy of a moving object.
speed: The rate at which an object moves.
velocity: The speed and direction of a moving object.
Assemble all the materials needed for the activity.
Write E = mc^{2} on the board and ask students what
the three letters in the equation represent. Emphasize that c stands for
a particular constant speed or velocity, that of light in a vacuum.
Demonstrate the parts of the apparatus students will use to find the
relationship between E and v (a replacement in the model for
c). Marbles must be dropped from rest, and the depth to which the
marbles penetrate must be measured with a dowel or skewer marked off in
centimeters. The depth should be measured from the top of the marble to the top
of the flour. The cup must be picked up, and the skewer must be viewed from the
side to measure depth accurately. Students should add a half centimeter to
their measurements in order to measure to the marble's center of mass.
Organize students into teams. Distribute the student handouts and materials.
Assign teams to one of the four heights from which to drop marbles. To ensure
data reliability, have several teams perform the same measurements multiple
times and average the results. As a class data table is going to be made, the
more data points the better.
Have students place newspaper on the floor, then place their cups filled
with flour in the center of a plastic pan. If students run out of clear area in
which to drop the marble, have them scoop out the marble(s) with the plastic
spoon, use the spoon to refill the cup, tap the base of the cup three times to
remove air pockets, and then use the dowel to level the flour to the cup's rim.
If students' fingers plunge in to retrieve marbles, they will pack down the
flour and their next set of data will be skewed with lower penetrating
distances. Results will also be skewed if they leave air pockets in the
flour.
Ask students how they might find the velocity of the marble as it hits the
flour. When they arrive at the correct mathematical strategy
have students calculate the velocity values for the four
given heights. (Students will need to convert centimeter drop heights to
meters.) Students may notice that doubling the height from which they drop the
marble does not double the velocity. If the calculations are too rigorous for
students, provide them with the values for velocity (see Activity Answer) and
tell students to square them for the v^{2} column of their
table.
When student teams are finished, create a class data chart on the board and
have students fill it in or have them enter their data into a computer
spreadsheet.
This is an ideal time to do some data analysis and statistics. Student
answers may vary quite a bit. For a given distance, ask students which of the
data points are "wrong" and which ones are "right." Discuss the best way to
average the numbers so students can graph just one depth for each distance
dropped. Cross out the two highest and lowest points (outliers) and average the
rest. Have students average all teams' depth results to determine final class
averages for each depth. Have students enter the results in the depth column in
the "Velocity vs. Energy Data Table" on their "Data Sheet" handout.
Discuss factors that may cause data to vary, i.e., nonuniform density of
the flour, problems with measuring depth, variations in tick marks, etc.
Emphasize that while accuracy is important, measurements may include a degree
of error. The goal is to see the pattern the data set (that energy is
proportional to velocity squared, not velocity).
Tell students they will make two graphs, the first with the yaxis
labeled Velocity (meters/second) and the xaxis labeled
Energy
(depth
in centimeters). The second graph will have the yaxis labeled
Velocity^{2} (meters^{2}/second^{2}) and the
xaxis labeled Energy (depth in centimeters). Meters are used on
the velocity axes to simplify graphing. Note that 0,0 must be used as a
data point when drawing the curve.
Have students plot the first graph. When students have finished plotting
points, review how to interpolate and draw a curve through a set of points
instead of drawing "dot to dot," as students will often do. Discuss the fact
that a curve shows that the two variables are not directly proportional.
Now have students plot their second graph in which the velocity is squared.
After students finish their second graph, help them draw a straight line
through as many points as possible. They should try to have roughly the same
number of points on either side of their lines. Then have student teams answer
the questions on their "Data Sheet" handout. Review answers as a class. What is
the most noticeable difference between the two graphs?
As an extension, have students use
to
calculate from what heights marbles would need to be dropped to double the
velocity for each height (beginning with the 10centimeter height). Have
the students repeat the experiment at this new set of heights (you may need to
use very widemouth cups in order for students to hit the target flour). Extend
the heights above the flour to 2 meters. Stop taking data when the marble hits
the bottom of the cup. Direct students to plot this new set of data, compare it
to their previous graphs, and find the slopes for each line. Are they the same?
Have students explain their results.
Marbles dropped from different heights accelerate toward the surface of the
flour, increasing their velocity and kinetic energy as they fall. The kinetic
energy that the marble has gained is then transferred to the flour as it
plunges in. The depth that the marble reaches in the flour is a measure of the
kinetic energy that is transferred to the flour (the energy deforms the flour
and makes the marble crater).
Student results may vary due to differences in flour density and errors in
dropping the marble from prescribed heights. When reviewing the table with
students, it would be best to eliminate the two lowest and highest values for
each height and average the rest. The summary data plotted should reveal that
energy is proportional to velocity squared.
Sample Results
Velocity vs. Energy Data Table
Distance (cm)

v (m/sec)

v^{2} (m^{2}/sec^{2})

Depth (cm)

0

0

0

0

10

1.4

2.0

0.5

25

2.2

4.9

1.3

50

3.1

9.8

2.4

100

4.4

19.6

4.1

The average error of the depth data was about +/ .5 cm.
Student Handout Questions
What is the shape of your Velocity vs. Energy graph? The shape of the
graph is a curve.
What is the shape of your
Velocity^{2}
vs. Energy graph? The shape of the graph should be close to a straight
line.
If a straight line on a graph indicates a direct relationship, is energy
(measured by depth) directly proportional to velocity or velocity squared?
The Velocity^{2} graph appears to show that energy is proportional to
velocity squared, not velocity.
Explain why Albert Einstein wrote E = mc^{2} instead of E
= mc. Students may forget that the c is simply the velocity of
light. Einstein wrote c^{2} because E is directly
proportional to velocity squared, not velocity.
Web Sites
NOVA—Einstein's Big Idea
www.pbs.org/nova/einstein
Hear top physicists explain E = mc^{2}, discover the legacy of
the equation, see how much energy matter contains, learn how today's physicists
are working with the equation, read quotes from Einstein, and more on this
companion Web site.
Answers from Scientists
www.skirball.org/exhibit/einstein_answers_light.asp
Answers several questions related to light and E = mc^{2}.
The Electromagnetic Spectrum
imagers.gsfc.nasa.gov/ems/waves3.html
Describes the electromagnetic spectrum and includes information on visible
light.
The Ultimate Physics Resource Site
serendip.brynmawr.edu/local/IIT/projects/Glasser.html
Includes physics links and activities.
Books
40 LowWaste, LowRisk Chemistry Experiments
by David Dougan. Walch Publishing, 1997.
Includes introductory labs on measurement, density, temperature, relative mass,
and more.
Energy by Jack Challoner. Dorling Kindersley, 1993.
Surveys various sources of energy and the ways in which they have been
harnessed.
Light
by David Burnie. Dorling Kindersley, 1999.
Explains many aspects of visible light and other forms of electromagnetic
energy.
Stop Faking It!: Light
by William C. Robertson. NSTA Press, 2003.
Provides information and activities to help teachers and students understand
light.
The "Messing With Mass" activity aligns with the following National Science
Education Standards (see books.nap.edu/html/nses)
and Principles and Standards for School Mathematics (see
standards.nctm.org/document/index.htm).
Grades 58
Science Standard
Physical Science
Mathematics Standard
Measurement
Grades 912
Science Standard
Physical Science
 Conservation of energy and the increase in disorder
Mathematics Standard
Measurement
Classroom Activity Author
Jeff
Lockwood taught high school astronomy, physics, and Earth science for 28 years.
He has authored numerous curriculum projects and has provided instruction on
curriculum development and science teaching methods for more than a decade.

The need for c
c
is necessary in Einstein's equation because whenever part of any piece of
matter is converted to pure energy, the resulting energy is by definition
moving at the speed of light. Pure energy is electromagnetic
radiation—whether light or Xrays or whatever—and electromagnetic
radiation travels at a constant speed of about 300,000 kilometers per
second.
The speed of light must be squared because of the nature of energy. When
something is moving four times as fast as something else, it doesn't have four
times the energy but rather 16 times the energy—in other words, that
figure is squared. So the speed of light squared is the conversion factor that
determines just how much energy lies within any type of matter.

