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Theme A — Space, Time and Motion

A.5.5 Velocity Addition Transformations — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 57 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

A.5.5 Velocity Addition Transformations — IB PHYSICS HL notes: Use relativistic velocity addition for speeds close to the speed of light.Galilean velocity addition is valid for velocities much less than c.To transform relativistic velocities, Lorentz transformations are used.It i…

  • Signs in the numerator and denominator match each other
  • The equation for u' is in the data booklet
  • Galilean limit
  • The result never exceeds c
  • Give relativistic answers in terms of c
  • Be careful about which frame the velocity is to be found in

Part 1 of 6Why Galilean addition fails

1
Galilean velocity addition is valid for velocities much less than c.
2
To transform relativistic velocities, Lorentz transformations are used.
3
It is needed when some velocities in the scenario are close to c.
4
Relativistic just means close to the speed of light.

Part 2 of 6Definitions of u, u' and v

u
u velocity of the object measured in the stationary frame S.
u'
u' velocity of the object measured in the moving frame S'.
v
v velocity of the moving frame S'.
c
c the speed of light.
Example
  • Example a missile fired ahead from a rocket: Person F in the rocket measures u', Person E on Earth measures u.

Part 3 of 6Relativistic velocity addition

Relativistic velocity addition
u'
(u − v) / (1 − uv/c2)
u
(u' + v) / (1 + u'v/c2)
Signs in the numerator and denominator match each other
  • Signs in the numerator and denominator match each other minus with minus for u', plus with plus for u.
The equation for u' is in the data booklet
  • The equation for u' is in the data booklet the one for u is obtained by changing the sign of v.
Galilean limit
  • Galilean limit u = u' + v when v and u' are much smaller than c.

Part 4 of 6Consequences

1The result never exceeds c
  • The result never exceeds c nothing can travel faster than light.
  • A rocket at 0.7c firing a probe at 0.5c gives (0.7 + 0.5)/(1 + 0.35) = 0.89c.
2If u'
If u' c then u = c for any v — light has the same speed for all observers.
3Give relativistic answers in terms of c
  • Give relativistic answers in terms of c c2 cancels in the denominator if u and v are in terms of c.

Part 5 of 6Using the equation

1
Decide which frame measures which velocity and which is the moving frame.
2
Write the sign of each velocity along the positive direction (the direction of v).
3
Choose the equation for u or u' and substitute.
4
Check the answer is less than c.
1Be careful about which frame the velocity is to be found in
  • Be careful about which frame the velocity is to be found in it decides whether u or u' is needed.
2Watch directions
  • Watch directions an object moving opposite to v has the opposite sign.

Part 6 of 6More from the specification

More from the specification
Relativistic velocity addition for muons
  • Relativistic velocity addition for muons adding relative reference velocities using Lorentz transformations ensures muon velocities never exceed vacuum light speed.
Inertial frames
  • Inertial frames reference frames moving at constant velocity relative to each other in which physics laws hold
Proper length and proper time
  • Proper length and proper time measured in the rest frame, with length contraction and time dilation seen by an observer in relative motion
Velocity addition
  • Velocity addition in special relativity relative velocities do not simply add, so no speed exceeds the speed of light
Events and intervals
  • Events and intervals a train and platform observers assign different distances and times to the same event using the Lorentz transformations
Reference frame
  • Coordinate system used by an observer.
Length contraction
  • Shortening of lengths measured along the direction of relative motion.
Proper length
  • Length measured in the rest frame of the object.
Constant velocity
  • Zero acceleration.
Space-time interval
  • Invariant quantity (cΔt)2 − (Δx)2.
Length
  • Resistance is directly proportional to the length of a wire.
Distance
  • Total length of path travelled, a scalar.
Gradient
  • Slope of a graph line.

Key terms

TermMeaning
Relativistic velocity additionTransformation of velocities between frames using the Lorentz equations.
Stationary frame SFrame in which the observer is at rest.
Moving frame S'Frame moving at velocity v relative to S.
Relativistic speedSpeed close to c.
uVelocity of the object in S.
u'Velocity of the object in S'.

Avoid these mistakes

1
Using Galilean addition for velocities near c.
2
Using the wrong sign in the denominator.
3
Mixing up u and u' — check which frame each is measured in.
4
Getting a result above c.
5
Entering c as a number when v and u are given as fractions of c.

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v = u + at  |  s = ut + ½at2  |  v2 = u2 + 2as  |  s = ½(u + v)t

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