1. Measuring relative speed
© D Hoult 2011

2. Imagine two observers, A and B in space ships moving with high speed

3. Imagine two observers, A and B in space ships moving with high speed relative to each other

4. Imagine two observers, A and B in space ships moving with high speed relative to each other
A decides that he / she would like to measure their relative speed

5. Imagine two observers, A and B in space ships moving with high speed relative to each other
A decides that he / she would like to measure their relative speed
How can this be done ?

6. Imagine two observers, A and B in space ships moving with high speed relative to each other
A decides that he / she would like to measure their relative speed
How can this be done ?
It is difficult to imagine a method which does not involve the use of

7. Imagine two observers, A and B in space ships moving with high speed relative to each other
A decides that he / she would like to measure their relative speed
How can this be done ?
It is difficult to imagine a method which does not involve the use of light or some other electro-magnetic radiation

8. Using pulses of light to measure the velocity of B relative to A

10. very close together, the 1st pulse of light takes (virtually) no time to get from A to B

11. both observers set clocks to zero using this pulse of light

12. A and B are much further apart when a 2nd pulse is sent

13. 2nd pulse of light reaches B

15. the next pulse of light has further to go to reach B

16. and so on…

17. each pulse of light has to go to x further to reach B than the previous pulse

18. For this reason, the time interval between the pulses received by B is

19. For this reason, the time interval between the pulses received by B is longer than the time interval between the pulses sent by A

20. Let B hold a mirror to reflect A’s pulses

21. The 2nd pulse arrives at the mirror

22. The 2nd pulse arrives at the mirror
remember pulse 1 was used to set A’s and B’s clocks to zero

23. At what time on A’s clock did the pulse reach the mirror ?

24. At what time on A’s clock did the pulse reach the mirror ?
Also, what was the distance between A and B at the instant when the light was reflected ?

26. If A waits to see the reflection, then t is…

27. If A waits to see the reflection, then t is…
the time half way between the time the pulse was sent and the time its reflection was received

28. and the distance is

29. and the distance is the distance light can travel in that time

30. We now define a constant (for a given motion), k such that

31. We now define a constant (for a given motion), k such that
time interval between received pulses
k =
time interval between transmitted pulses

32. We now define a constant (for a given motion), k such that
time interval between received pulses
k =
time interval between transmitted pulses TR
k = TT

33. We now define a constant (for a given motion), k such that
time interval between received pulses
k =
time interval between transmitted pulses TR
k = TT
obviously, the value of k depends on

34. We now define a constant (for a given motion), k such that
time interval between received pulses
k =
time interval between transmitted pulses TR
k = TT
obviously, the value of k depends on the relative speed of transmitter and receiver; a higher speed gives a

35. We now define a constant (for a given motion), k such that
time interval between received pulses
k =
time interval between transmitted pulses TR
k = TT
obviously, the value of k depends on the relative speed of transmitter and receiver; a higher speed gives a higher value of k

36. At the instant when A and B are together, the first pulse of light is sent

37. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero

38. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero
The 2nd pulse is sent by A at time t and therefore received by B at time

39. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero
The 2nd pulse is sent by A at time t and therefore received by B at time k t

40. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero
The 2nd pulse is sent by A at time t and therefore received by B at time k t
Any reflected pulses will have the same factor k relating the time interval between received and transmitted pulses

41. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero
The 2nd pulse is sent by A at time t and therefore received by B at time k t
Therefore the 2nd pulse reflection is received by A at time

42. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero
The 2nd pulse is sent by A at time t and therefore received by B at time k t
Therefore the 2nd pulse reflection is received by A at time k2 t

43. At the instant when A and B are together, the first pulse of light is sent
Both clocks are set to zero
The 2nd pulse is sent by A at time t and therefore received by B at time k t
Therefore the 2nd pulse reflection is received by A at time k2 t
A knows that the time of reflection is half way between t and k2 t

44. Time on A’s clock at the instant when reflection occurred is

45. Time on A’s clock at the instant when reflection occurred is
t + k2 t 2

46. Time on A’s clock at the instant when reflection occurred is
t + k2 t 2
The light went from A to B and back in

47. Time on A’s clock at the instant when reflection occurred is
t + k2 t 2
The light went from A to B and back in
k2 t - t

48. Time on A’s clock at the instant when reflection occurred is
t + k2 t 2
The light went from A to B and back in
k2 t - t
So the distance between A and B when the reflection occurred was

49. Time on A’s clock at the instant when reflection occurred is
t + k2 t 2
The light went from A to B and back in
k2 t - t
So the distance between A and B when the reflection occurred was
(k2 t - t) c 2

50. v =

51. distance
v =
time

52. distance
v =
time
therefore
v =

53. distance
v =
time
therefore
(k2 t - t) c 2
v =
t + k2 t 2

54. distance
v =
time
therefore
(k2 t - t) c 2
v =
t + k2 t 2
which simplifies to…

55. (k2 - 1) c
v =
(k2 + 1)

56. All A has to do now is find the value of the constant k for this motion…

57. This can be done by simply measuring the

58. This can be done by simply measuring the time interval between reflected pulses and using the fact that

59. This can be done by simply measuring the time interval between reflected pulses and using the fact that
time interval between reflected pulses
time interval between transmitted pulses

60. This can be done by simply measuring the time interval between reflected pulses and using the fact that
time interval between reflected pulses =
time interval between transmitted pulses

61. This can be done by simply measuring the time interval between reflected pulses and using the fact that
time interval between reflected pulses
= k2
time interval between transmitted pulses