Mass and Energy

Beta Factor

vu /

Relativity matters when speeds are close to that of light.• v > 0.1c• v/c > 0.1 (less than 1% error)

The beta factor is often used in relativity to scale values according to the speed of light.

21

1

Total Energy

Total energy is based on kinetic energy.• No potential, only kinetic• Classically E = ½ mv2

With relativity the total energy depends on β.

Check this definition at low speed.• Approximation:

• Application:

2

2

2

1mc

mcE

21

1

12

x

x

22

2

2

2

21

1mc

c

vmc

22

2

2

2

1

1mvmc

mc

Low Energy Match

To match kinetic energy with the result at low velocity, the extra term must be subtracted.

The kinetic energy also depends on β.

222

2

1mvmcmcE

2

22

2

1 mcK

mcmcK

mcEK

Rest Energy

The extra term for energy is still there when an object is at rest.

This energy only depends on the mass.

22

2 11/1

0/

0

mcmcE

cv

v

Energy from Mass The rest energy exists

in an object’s inertial frame.• No velocity• No potential• Just mass

Conservation of energy implies that this rest energy can be converted to other forms.

20 mcE

Hot House

An average house uses 2 x 1010 J for heating and cooling in one year.

What mass equivalent is that?

Solve for the rest mass.

Use c = 3 x 108 m/s.m = 2 x 10-7 kg

• About 1/10 the mass of a human hair

2

2

/ cEm

mcE

Electron Gun

An electron is accelerated through a potential of 106 V.• Rest energy 0.511

MeV

What is the final speed in units of c?

The rest energy doesn’t change.• Potential to kinetic

γ = 3.0, v = 0.94c

1

1

2

2

cm

eV

cmeV

e

f

ef