Einstein's Big Idea
Squaring Off With Velocity
Students investigate the meaning of c2 in E =
mc2 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.
Students will be able to:
explain what the c2 in E = mc2
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 v2, not v.
- copy of the "Squaring Off With Velocity" student handout
- copy of the "Data Sheet" student handout
- 1 lb flour in a plastic bag
- plastic pan (about shoebox size)
- two 8-oz plastic cups
- four standard-sized (about 1 cm) glass marbles
- plastic ruler
- meter stick
- plastic spoon
- wood dowel or wood barbecue skewer
- felt-tip pen
- graph paper
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
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 = mc2.
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 used-10, 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/sec2), and d = the distance the marble falls (in meters)
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 = mc2 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
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 v2 column of their
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
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., non-uniform 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 y-axis
labeled Velocity (meters/second) and the x-axis labeled
in centimeters). The second graph will have the y-axis labeled
Velocity2 (meters2/second2) and the
x-axis 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
calculate from what heights marbles would need to be dropped to double the
velocity for each height (beginning with the 10-centimeter height). Have
the students repeat the experiment at this new set of heights (you may need to
use very wide-mouth 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.
Velocity vs. Energy Data Table
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
vs. Energy graph? The shape of the graph should be close to a straight
If a straight line on a graph indicates a direct relationship, is energy
(measured by depth) directly proportional to velocity or velocity squared?
The Velocity2 graph appears to show that energy is proportional to
velocity squared, not velocity.
Explain why Albert Einstein wrote E = mc2 instead of E
= mc. Students may forget that the c is simply the velocity of
light. Einstein wrote c2 because E is directly
proportional to velocity squared, not velocity.
NOVA—Einstein's Big Idea
Hear top physicists explain E = mc2, 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
Answers several questions related to light and E = mc2.
The Electromagnetic Spectrum
Describes the electromagnetic spectrum and includes information on visible
The Ultimate Physics Resource Site
Includes physics links and activities.
40 Low-Waste, Low-Risk Chemistry Experiments
by David Dougan. Walch Publishing, 1997.
Includes introductory labs on measurement, density, temperature, relative mass,
Energy by Jack Challoner. Dorling Kindersley, 1993.
Surveys various sources of energy and the ways in which they have been
by David Burnie. Dorling Kindersley, 1999.
Explains many aspects of visible light and other forms of electromagnetic
Stop Faking It!: Light
by William C. Robertson. NSTA Press, 2003.
Provides information and activities to help teachers and students understand
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
- Conservation of energy and the increase in disorder
Classroom Activity Author
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.