This ppt explains about time dilation, variation of mass with velocity, mass-energy equivalence

1. TIME DILATION
Dr.R.Hepzi Pramila Devamani,
Assistant Professor of Physics,
V.V.Vanniaperumal College for Women,
Virudhunagar

2. TIME DILATION
• Derivation
• Imagine a gun placed at the position (x' y z) in S' (Fig. 5.7).
• Suppose it fires two shots at times t1’ and t2’ measured with
respect to S'.
• In S' the clock is at rest relative to the observer. The time
interval measured by a clock at rest relative to the observer
is called the proper time interval.
• Hence, to = t2’ –t1’ is the time interval between the two
shots for the observer in S'.
• Since the gun is fixed in S', it has a velocity v with respect
S in the direction of the positive X-axis.
• Let t = t2 – t1 represent the time interval between the two
shots as measured by an observer in S.

3. TIME DILATION

4. TIME DILATION
• From inverse Lorentz transformations we have

5. TIME DILATION
• Thus, the time interval, between two events
occurring at a given point in the moving frame
S' appears to be longer to the observer in the
stationary frame S.
• Thus a stationary clock measures a longer time
interval between events occurring in a moving
frame of reference than does a clock in the
moving frame. This effect is called time
dilation.

6. Explanation
• Time interval, like a length interval, is not
absolute. It is also affected by relative motion.
• The time-interval between two ticks as judged
from moving frame S' is to.
• Suppose another observer measures the time
interval between same two ticks as t, from a
stationary frame S, relative to which the clock is
moving with a speed v. Then,
t =
𝑡0
1 −
𝑣2
𝐶
2

7. Explanation
• This equation shows that to the observer in
frame S, the time appears to be increased by a
factor 1 −
𝑣2
𝑐2
• Thus a stationary clock measures a longer
time-interval between events occurring in a
moving frame of reference than does a clock
in the moving frame.

8. Explanation
• In other words, a moving clock appears to
be slowed down to a stationary observer.
This effect is called time dilation.
• If v = c, t = ∞, It means that a clock
moving with the speed of light will
appear to be completely stopped to a
stationary observer.

9. The Twin Paradox
• Consider two exactly identical twin brothers. Let
one of the twins go to a long space journey at a
high speed in a rocket and the other stay behind
on the earth.
• The clock in the moving rocket will appear to go
slower than the clock on the surface of earth.
• Therefore, when he returns back to the earth, he
will find himself younger than the twin who
stayed behind on the earth!

10. Variation of Mass with Velocity
Derivation
• Consider two systems S and S' S' is moving with a constant velocity
V relative to the system S, in the positive X-direction (F1g. 5.8).
• Suppose that in the system S', two exactly similar elastic balls A and
B approach each other at equal speeds (i.e., u and -u).
• Let the mass of each ball be m in S’. They collide with each other
and after collision coalesce into one body.
• According to the law of conservation of momentum.
• Momentum of ball A+ momentum of ball B= momentum of
coalesced mass.
• mu + (-mu)= momentum of coalesced mass= 0.
• Thus the coalesced mass must be at rest in S’ system.

11. Variation of Mass with Velocity

12. Variation of Mass with Velocity
• Let us now consider the collision with reference to the system S.
• Let u1 and u2 be the velocities of the balls relative to S. Then,
• After Collision, velocity of the coalesced mass is v relative to the
system

13. Variation of Mass with Velocity
• Let mass of the ball A travelling
with velocity u1 be m1 and that of
B with velocity u2 be m2 in the
system S.
• Total momentum of the balls is
conserved.

14. Variation of Mass with Velocity

15. Variation of Mass with Velocity

16. Variation of Mass with Velocity

17. Explanation
• In Newtonian mechanics, the mass of a body is taken constant
and independent of velocity.
• But according to special theory of relativity, the mass of a
body changes with its velocity.
• That is, the mass of a body in motion is different from the
mass of the body at rest.
• The mass of a body varies with its velocity according to the
relation
• m =
𝑚0
1 −
𝑣2
𝐶2
• Here, m0 is the "rest mass" of the body, c is the velocity of
light and v is the velocity of the body.

18. Explanation
• Following conclusions are drawn from mass variation formula
• As the velocity v of the particle relative to the observer increases,
the mass of the particle increases.
• As v → 𝑐, 𝑚 → ∞. This means that no material particle can
have a velocity equal to, or greater than, the velocity of light.
• When v << c, then v2 / c2 can be neglected as compared to 1 and
so m ≈ m0.
• This means that at ordinary velocities the difference
between m and m0 is so small, that it can not be detected.
• The increase in mass has been directly verified in various
electron diffraction experiments and also in the operation of
particle accelerators

19. Mass Energy Equivalence
• Force is defined as rate of change of
momentum
• F =
𝑑
𝑑𝑡
(mv) (1)
• According to the theory of relativity, both mass
and velocity are variable.
• F =
𝑑
𝑑𝑡
(mv) = m
𝑑𝑣
𝑑𝑡
dx + v
𝑑𝑚
𝑑𝑡
dx (2)

20. Mass Energy Equivalence
• Let the force F displace the body through a
distance dx.
• Increase in the kinetic energy (dEk) of the body is
equal to the work done (F dx).
• Hence, dEk = F dx = m
𝑑𝑣
𝑑𝑡
dx + v
𝑑𝑚
𝑑𝑡
dx
• Or dEk = mv dv + v2dm (3)
• According to the law of variation of mass with
velocity, m =
𝑚0
1 −
𝑣2
𝐶
2
(4)

21. Mass Energy Equivalence
• squaring both sides, m2 = m0
2/ (1-v2 /c2)
• m2 c2 = m0
2 c2 + m2v2
• Differentiating, c2 2m dm = m2 2vdv + v22mdm
• c2 dm = mvdv + v2dm (5)
• From Eqs. (3) and (5),
• dEk = c2 dm (6)
• Thus, a change in K.E., dEk is directly
proportional to a change in mass dm.

22. Mass Energy Equivalence
• When a body is at rest, its velocity is zero,
(K.E = 0) and m=m0, when its velocity is v, its
mass becomes m.

23. Mass Energy Equivalence
• This is the relativistic formula for KE.
• When the body is at rest, the internal energy
stored in the body is m0 c2.
• m0 c2 called the rest mass energy.
• The total energy (E) of the body is the sum of
KE(Ek ) and rest mass energy (m0 c2).
• E = Ek + m0 c2 = m c2 - m0 c2+ m0 c2 = m c2
• E = m c2 . This is Einstein's mass-energy relation.

24. Mass Energy Equivalence
• Explanation
• According to the theory of relativity, if the
mass m of a particle is completely converted
into energy , then E = m c2
• Here, c is the velocity of light.
• This is the famous Einstein's mass-energy
relation

25. Mass Energy Equivalence
• It states that an energy m c2 associated with a mass m.
• Conversely, a mass E / c2 associated with energy.
• This relation states a universal equivalence between
• mass and energy. It means that mass may appear as energy
as mass.
• The relationship (E = m c2) between energy and mass forms
the basis of understanding nuclear reactions such as fission
and fusion.
• These reaction take place in nuclear bombs and reactors.
• When uranium nucleus is split up, the decrease in its total
rest mass appears in the form of an equivalent amount of
K.E, of its fragments.