General Physics 2 Activity Sheet
Quarter 4 – MELC 24
Week 7
Rest Energy, Relativistic Energy and
Momentum
REGION VI – WESTERN VISAYAS
General Physics 2
Activity Sheet -Relativistic Energy and Speed of Light
First Edition, 2021
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Introductory Message
Welcome to General Physics 2
The Learning Activity Sheet is a product of the collaborative efforts of the Schools
Division of Sagay City and DepEd Regional Office VI - Western Visayas through the
Curriculum and Learning Management Division (CLMD). This is developed to guide the
learning facilitators (teachers, parents and responsible adults) in helping the learners meet
the standards set by the K to 12 Basic Education Curriculum.
The Learning Activity Sheet is self-directed instructional materials aimed to guide
the learners in accomplishing activities at their own pace and time using the contextualized
resources in the community. This will also assist the learners in acquiring the lifelong learning
skills, knowledge and attitudes for productivity and employment.
For learning facilitator:
The General Physics 2 Activity Sheet will help you facilitate the leaching-learning
activities specified in each Most Essential Learning Competency (MELC) with minimal or no
face-to-face encounter between you and learner. This will be made available to the learners
with the references/links to ease the independent learning.
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The General Physics 2 Activity Sheet is developed to help you continue learning
even if you are not in school. This learning material provides you with meaningful and
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understand the instructions then perform the activities and answer the assessments. This will
be returned to your facilitator on the agreed schedule.
Name of Learner: _________________________________________________________
Grade and Section: __________________________ Date: ________________________
GENERAL PHYSICS 2 ACTIVITY SHEET
Rest Energy, Relativistic Kinetic Energy and Momentum
I. Learning Competency with Code
1. Calculate kinetic energy, rest energy, momentum, and speed of objects moving with speeds
comparable to the speed of light Week 9 STEM_GP12MPIVg-42
II. Background Information for Learners
In Einstein’s theory of relativity, matter can be considered a form of energy and can be
converted into energy. Furthermore, energy can also be converted into matter. The equivalence
between matter and energy is given by his famous equation, E = mc2.
● REST ENERGY (E0) – the energy equivalent to the mass of a particle at rest in an inertial frame of
reference.
where: E – rest energy (unit: joules)
E0 = mc2
m – rest mass of the particle (unit: kilogram)
c – speed of light in vacuum(constant value: 3x108 m/s)
An atomic bomb uses nuclear fission
(splitting) of Uranium-235 element. In
Figure 1, the U-235 is struck by a neutron
and it causes the unstable U-236 to split
into two lighter elements (Kr-92 and Ba-
141) and 3 neutrons. During this process,
the conservation of mass must be
observed, but according to experiments,
the total mass of K-92, Ba-141 and 3
neutrons is less than the mass of U-236.
This means that the missing mass (m =
3.56x10-28 kg) was converted into energy
(E = 3.20×10−11 J).
Now, put this in a bigger scale as shown in
Figure 2. In August 6, 1945, the United
States of America dropped a Uranium
Bomb in Hiroshima and it uses nuclear
fission on 0.88kg of Uranium. During its
nuclear fission, 0.0008 kg of mass was lost
and that amount of mass was converted
into an enormous energy of 72x1012 J that
destroyed the whole city.
Sample Problem: How much energy is stored in a 1 gram of mass.
Given: Solution: E0 = mc2
m = 1g or 0.001kg E0 = (0.001kg) (3 x108 m/s)2
8
c = 3 x10 m/s E0 = 90,000,000,000,000 J or
E0 = 90 x 1012 J
● RELATIVISTIC KINETIC ENERGY (KE) – the energy of a mass moving at very high speed.
The total energy is the summation of rest energy (E0) and the kinetic energy (KE) as shown below:
Etotal = KE + E0
Therefore, you get the kinetic energy (KE) by subtracting the rest energy (E0) from the total energy
(Etotal): KE = Etotal – E0
By substituting Etotal and E0 with its equivalent equations, we get the relativistic kinetic energy as:
where: KE – relativistic kinetic energy (unit: joules)
KE = γmc2 – mc2 γ – Lorentz factor (unitless)
or m – rest mass of the particle (unit: kilogram)
KE = mc2 (γ – 1) c – speed of light in vacuum (constant value: 3x108 m/s)
In your old television set (CRT TV), pictures
are displayed by bombarding electrons on
the screen as shown in Figure 3. Those
electrons came from the cathode ray located
at the back portion of the tube. The
deflection coils are responsible in deflecting
each passing electron to a specific target
spot on the viewing screen. When the
electrons hit the viewing screen, the screen
begins to light up and creates the picture
frames that we see. Electrons coming out of
the cathode can travel at speeds of 0.1c or
30,000,000 m/s. Using the formula for
relativistic KE, each electron with a mass of
9.11x10-31 kg has a relativistic KE of
4.13x10-16 joules.
Sample Problem: What will be the relativistic KE of the 1 gram particle moving at a speed of 0.7c?
Given: 1 1 1
Solution: Lorentz factor: ϒ= = 2
= 1.4
2 2
m = 1g or 0.001kg √1− (𝑣) √1− (0.7𝑐) √1− (0.7)
𝑐 𝑐
v = 0.7c or 2.1x108 m/s
Relativistic KE: KE = mc (γ – 1)
2
c = 3 x108 m/s
KE = (0.001kg) (3x108m/s)2 (1.4 – 1)
KE = 120,000 J or
KE = 120 x 103 J
● RELATIVISTIC MOMENTUM (p) – the particle’s resistance to stopping given that the particle is
moving at very high speed. The relativistic momentum approaches an infinite value as v approaches
the value of c and it is reduced to its classical definition (p = mv) when v is much less than the value
of c. where: p – relativistic momentum (unit: kg·m/s)
γ – Lorentz factor (unitless)
p = γmv m – mass of the particle (unit: kilogram)
v – velocity of the particle (unit: m/s)
Sample Problem: What will be the relativistic momentum of a 1 gram particle moving at a speed of 0.7c?
1 1
Given: Solution: Lorentz factor: ϒ= = = 1.4
𝑣 2 0.7𝑐 2
m = 1g or 0.001kg √ 1− ( ) √ 1− ( )
𝑐 𝑐
v = 0.7c or 2.1x108 m/s Relativistic p: p = γmv
c = 3 x108 m/s p = (1.4) (0.001kg) [(0.7)(3x108m/s)]
p = (1.4) (0.001kg) [2.1x108m/s]
p = 294,000 kg·m/s or
p = 294 x 103 kg·m/s
III. Activity Proper
A. An electron has a rest mass of 9.11 x10-31 kg. Calculate the following:
(a) Rest energy
(b) Relativistic kinetic energy if it moves at 0.6c
(c) Relativistic momentum if it moves at 0.6c
B. Directions: Answer the following in your answer sheet.
1. Describe the value of Lorentz Factor if the speed of the object increases and approaches the value
of c.
_______________________________________________________________________________
_______________________________________________________________________________
2. Looking at the formula of Lorentz Factor, is it possible that the object’s speed (v) is equal to c?
_______________________________________________________________________________
_______________________________________________________________________________
3. Describe the value of Lorentz Factor if the speed of the object is reduced to 1,000,000 m/s or lower.
_______________________________________________________________________________
_______________________________________________________________________________
IV. Reflection
Is it possible for a spaceship travels at the speed of light? Why?
_______________________________________________________________________________
_______________________________________________________________________________
V. Key Answers
A. Given: m = 9.11 x10-31 kg v = 0.6c or 2.7x108 m/s c = 3 x108 m/s
2
(a) Rest Energy E0 = mc
E0 = (9.11 x10-31 kg) (3 x108 m/s) 2
E0 = 8.2 x 10-14 J
1 1 1
(b) Lorentz factor: ϒ= = = = 1.25
√1− (v)
2
√1− (0.6c)
2 √1− (0.6)2
c c
Relativistic KE: KE = mc2 (γ – 1)
KE = (9.11 x10-31kg) (3x108m/s)2 (1.25 – 1)
KE = 2.05 x10-14 J
(c) Relativistic momentum: p = γmv
p = (1.25) (0.001kg) (2.7x108m/s)
p = 3.4 x105 kg·m/s
Guide Questions:
1. The Lorentz Factor will also increase starting from a value of 1 up to a value approaching infinity.
2. NO, because when the object’s speed is the equal to the speed of light, the value for the Lorentz Factor
becomes infinite. Theoritically, this only tells us that no object with mass can travel at a speed of light – perhaps
for now.
3. The value of Lorentz Factor becomes approximately 1. This puts the relativistic momentum back to its
classical formula (p = mv) and makes the relativistic KE unusable.
IV. Reflection
NO, because a spaceship has mass and it is impossible to anything that has mass to travel at a speed of light.
The only thing that travels at a speed of light are photons, which are massless.
VI. Links and other References
https://www.indiamart.com/proddetail/crt-tv-19412832712.html
https://electricalfundablog.com/cathode-ray-tube-crt/
https://app.emaze.com/@AITQOLOC#1
https://www.atomicheritage.org/history/bombings-hiroshima-and-nagasaki-1945
https://theconversation.com/world-politics-explainer-the-atomic-bombings-of-hiroshima-and-
nagasaki-100452
Young, H. & Freedman, R. (2012). University Physics with Modern Physics (13 th Edition), 1114-
1153. Retrieved from
https://www.academia.edu/39499005/Full_Book_University_Physics_13th_Edition_PDF_KD
Halliday, D., Resnick, R., & Walker, J. (2011). Fundamentals of Physics (9th Edition), 924-950.
Retrieved from
https://www.academia.edu/31441276/Fundamentals_of_Physics_Extended_9th_HQ_Hallida
y_pdf
