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Relativistic Mass, Energy, Momentum. Classical physics: p = mv. When consider relativity:. p = m 0 v/ √(1 – v 2 /c 2 ). m 0 = rest mass. Consider m = m 0 / √(1 – v 2 /c 2 ). mass as measured in a reference frame in which it moves at speed v. √(1 – v 2 /c 2 ).

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Classical physics: p = mv

When consider relativity:

p = m0v/ √(1 – v2/c2 )

m0 = rest mass

Consider m = m0 / √(1 – v2/c2 )

mass as measured in a reference frame

in which it moves at speed v


√(1 – v2/c2 )

As v  c, √(1 – v2/c2 )  0

m  ∞

To accelerate an object up to speed c

would require an infinite amount of energy

c is the speed limit of the universe

if v > c, imaginary mass (tachyon)


Relativistic Kinetic Energy

KE = mc2 – m0c2

m is the relativistic mass

m0 is the rest mass

m0c2 is the rest energy

for an object at rest, the equivalent amount of energy which can be produced from its mass E0 = m0c2


Total energy: E = mc2 = m0c2 + KE

Observe an increase in mass (m) due to

an increase in KE

If the energy of a system changes by ΔE,

the mass of the system changes by Δm:

ΔE = (Δm)c2


π0 meson (m0 = 2.4 x 10-28 kg) travels

at 0.80c

relativistic mass:

4.0 x 10-28 kg

total energy:

3.6 x 10-11 J

rest energy:

2.16 x 10-11 J

kinetic energy:

1.44 x 10-11 J


Consequences of Special Relativity

For an observer outside a moving

frame of reference:

length

mass

time

speed of light

decreases

increases

slows down

is constant


Albert Einstein (1905)

relationship between:

space and time

mass and energy

(E = mc2)


Space and time exist within

the universe

Space-time: 3 dimensions of space

1 dimension of time

Motion through space affects our

motion in time

As speed increases, time slows down


Δy’

y’

L

y

Δy

Δx’

Δx

x’

x

L = √(Δx)2 + (Δy)2

L = √(Δx’)2 + (Δy’)2


4-dimensional space-time interval

s = √(Δx)2 + (Δy)2 +(Δz)2 –(c Δt)2


In a different frame of reference, trade a

little space for a little time.


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