SME
University Physics III: Waves, Optics & Modern · Problem type
Relativistic energy and momentum
Use this for “A proton (rest energy 938.3 MeV) moves at 0.8c. Find its total energy, its kinetic energy and its momentum in MeV/c.”
Diagram
E
2
=
(pc)
2
+
(mc²)
2
:
E
=
√
0.3833
2
+
0.511
2
=
0.6388
MeV.
The
angle
at
the
origin:
cos
θ
=
pc
E
=
0.6
=
β.
Drawn to scale: mc² up from the end of pc, E closing the triangle.
This is an example. Type your own values to start a new problem.
γ
=
1
1
−
(
)
2
=
E (
)
=
×
mc² (
)
K
Kinetic energy
calculated
pc
Momentum times c
calculated
Formulas
γ
=
1
÷
√
1 −
β
2
1.25
=
1
÷
√
1 −
0.6
2
E
=
γ
×
mc²
0.6388 = 1.25 × 0.511
K
=
(
γ
−
1)
×
mc²
0.1278 = (1.25 − 1) × 0.511
pc
=
γ
×
β
×
mc²
0.3833 = 1.25 × 0.6 × 0.511
E
2
=
(pc)
2
+
(mc²)
2
0.6388
2
=
(0.3833)
2
+
(0.511)
2
Assumptions
•
Energies are in MeV and the momentum is given as pc, also in MeV (p in MeV/c); an electron’s rest energy mc² is 0.511 MeV, a proton’s 938.3 MeV.
•
With γ = 1 ÷ √(1 − β²), the total energy is E = γmc², the kinetic energy K = (γ − 1)mc² and the momentum pc = γβmc².
•
E, pc and mc² make a right triangle, E² = (pc)² + (mc²)², the same in every frame; at rest E = mc² and pc = 0.
Step-by-step
WE KNOW
mc² = 0.511 MeV, β = 0.6
FIND
Lorentz factor (γ), total energy (E), kinetic energy (K), momentum times c (pc)
Step 1 · Find Lorentz factor (γ)
Use γ = 1 ÷ √(1 − β²)
The
Lorentz
factor
for
this
speed:
1
at
rest,
growing
without
limit
as
β
nears
1.
γ
=
1
÷
√
1 −
0.6
2
γ
=
1
÷
√
1 − 0.36
γ
=
1
÷
√
0.64
γ
=
1
0.8
γ
=
1.25
Step 2 · Find total energy (E)
Use E = γ × mc²
The
total
energy
is
γ
times
the
rest
energy.
E
=
1.25
×
0.511
E
=
0.6388
MeV
Step 3 · Find kinetic energy (K)
Use K = (γ − 1) × mc²
The
kinetic
energy
is
what
the
total
holds
beyond
the
rest
energy:
(
γ
−
1)
rest
energies.
K
=
(1.25
−
1)
×
0.511
K
=
0.25
×
0.511
K
=
0.1278
MeV
Step 4 · Find momentum times c (pc)
Use pc = γ × β × mc²
The momentum p = γmv; times c it is γβ rest energies.
pc = 1.25 × 0.6 × 0.511
pc = 0.75 × 0.511
pc = 0.3833 MeV
Check
Put the numbers back in. Both sides should match.
1.25
=
1
÷
√
1 −
0.6
2
✓
0.6388 = 1.25 × 0.511 ✓
0.1278 = (1.25 − 1) × 0.511 ✓
0.3833 = 1.25 × 0.6 × 0.511 ✓
0.6388
2
=
(0.3833)
2
+
(0.511)
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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