AIN SHAMS UNIVERSITY
Faculty of Engineering
Engineering Physics & Mathematics
Department
Modern Physics & Quantum Prof. Wael Fikry
Mechanics Dr. Michael Gad
Problem Set 1
Mass, Energy & Momentum in Special Relativity
Constants in SI units: Electron charge (e) = 1.6×10-19, Electron rest mass (mo) = 9.1×10-31,
Speed of light = 3×108, Planck’s constant (h) = 6.62×10-34
Part I: Problems
1. Calculate the velocity of an electron whose energy is 2 MeV
2. An electron is accelerated through a potential difference of 105 V
(a) What is the kinetic energy of the electron?
(b) Calculate the electron speed.
3. Calculate the mass of the electron when it is moving with a K.E. of 10 MeV.
4. Calculate the momentum of a 1 MeV electron.
5. A particle has a total energy of 6x103 MeV and a momentum of 3x103 MeV/c. What is its rest mass?
6. A particle of rest mass (mo = 2.4×10-28 kg) travels at a speed of (0.8c).
(a) Calculate its kinetic energy using the relativistic and the classical approaches.
(b) If this body is transformed by decay into electromagnetic radiation, how much energy is released.
7. The speed of a particle increases by a factor of (100) from (0.0098c) to (0.98c). By what factors do
its momentum and kinetic energy increase? Do your calculations using both the relativistic and the
classical approaches.
8. Two identical bodies, each with rest a mass (mo), approach each other with equal velocities (v)
collide, and stick together in a perfectly inelastic collision. Determine the rest mass of the composite
body.
MASS, ENERGY & MOMENTUM IN SPECIAL RELATIVITY 1/2
9. A particle of rest mass (mo) moving with a speed (v = 0.8c) makes a completely inelastic collision
with a particle of rest mass (3mo) that is initially at rest. What are the speed and the rest mass of the
resulting single body?
∂KE
10. Prove that the velocity of a particle is related to its KE and momentum by v = for any motion
∂p
described classically or relativistically.
11. Show that relativistic KE= (m – mo)c2 reduces to nonrelativistic KE = ½mov2 when v is very much
smaller than c. [(1- x)n≈ (1- nx) if x << 1]
12. Prove that a free electron at rest cannot capture or absorb a photon.
13. Solar energy reaches the earth at the rate of about 1.4 kW
per square meter of surface perpendicular to the direction of
the sun. By how much does the mass of the sun decrease per
second owing to this energy loss? The mean radius of the
earth’s orbit is 1.5×1011 m.
Part II: State whether the following statements are true or false, and if false state
the reason:
1. In modern physics, any value of the speed of a particle is possible.
2. As the speed of the particle increases, its rest mass increases.
3. A massless particle has zero momentum.
4. A particle moves at a speed (v = 0.5c). Its kinetic energy is found higher when calculated using
the relativistic relation than when using the classical relation.
5. Classical physics was sufficient to explain phenomena like the gravitational lens.
Part III: MATLAB exercise (Optional)
a) Write a Matlab code to plot three separate figures showing:
-the ratio of the relativistic mass to the rest mass of an electron (m / mo ) ,
-the electron momentum using the classical and the relativistic approaches, and
-the electron kinetic energy using the classical and the relativistic approaches
all versus the ratio of the electron speed to the speed of light 0 < v / c < 1
b) Solve problem (9) using Matlab.
MASS, ENERGY & MOMENTUM IN SPECIAL RELATIVITY 2/2
