3 Speed of light revisited

With the length of the second firmly tied to atomic physics, rather than the rotation of the Earth, surely time is now well and truly nailed down? Unfortunately, no it isn’t! To find out why, you will need to revisit the speed of light.

Suppose you’re a police officer wanting to use a radar or laser gun to measure the speed of traffic. It’s no good using the gun when you’re driving along, as that would only give you the relative speed between your car and the other cars on the road. To measure the speed of a suspect vehicle correctly, you must be standing motionless beside the road.

Watch this short video in which Marcus introduces relative speed, then have a go at the calculations yourself in Activity 1.

Download this video clip.Video player: Video 2

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Transcript: Video 2 Relative speed (part 1)

MARCUS DU SAUTOY

What is ‘relative speed’ and why does it matter? Well, imagine you’re a police officer with a speed gun, trying to detect speeding motorists. You park your police car at the roadside, and point the speed gun at the passing traffic. Bingo – you’ve got a reading of their speed. But what if the police car is moving too? Of course, we have to remember that all speeds are relative. And what matters here is what speed the suspect is doing relative to the roadside. In this case, the suspect car isn’t actually going at 150 kilometres per hour. That’s its speed relative to the police car. So, the police car’s speed and direction need to known in order to work out if the suspect car is actually speeding. And it’s a similar scenario if a car is driving away from us. This lines up exactly with our natural intuition. We know that if we hit something coming towards us fast, it’s likely to have much greater impact than if we hit something going in the same direction and at a similar speed. So the maths makes perfect sense.

End transcript: Video 2 Relative speed (part 1)

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Video 2 Relative speed (part 1)

Activity 1 Relative speeds

Timing: Allow about 10 minutes

Select the answer for

Question 1a

here

With that in mind, imagine the three scenarios below. What is the relative speed in each case?

(a) A stationary police car at the roadside observes a car travelling at 100 km/h towards it.

50 km/h

100 km/h

150 km/h

200 km/h

 

a. 

50 km/h

b. 

100 km/h

c. 

150 km/h

d. 

200 km/h

The correct answer is b.

Select the answer for

Question 1b

here

(b) A police car travelling at 50 km/h observes a car travelling at 100 km/h towards it.

50 km/h

100 km/h

150 km/h

200 km/h

 

a. 

50 km/h

b. 

100 km/h

c. 

150 km/h

d. 

200 km/h

The correct answer is c.

Select the answer for

Question 1c

here

(c) A police car travelling at 50 km/h observes a car travelling at 100 km/h away from it.

50 km/h

100 km/h

150 km/h

200 km/h

 

a. 

50 km/h

b. 

100 km/h

c. 

150 km/h

d. 

200 km/h

The correct answer is a.

Discussion

Relative speeds are calculated by adding the two speeds when they are in opposite directions and by subtracting when they are in the same direction. Therefore the relative speeds, as measured by the radar gun, are calculated as follows:

• a.0 + 100 = 100 km/h
• b.50 + 100 = 150 km/h
• c.100 - 50 = 50 km/h

These results line up with the intuition that a head-on collision has a much higher impact than hitting a car driving in the same direction. But only in the first scenario above does the relative speed match the actual speed of Car B.

Returning to the speed of light: where should we stand to measure it? Again, imagine three scenarios:

• a.A stationary physicist measures the speed of a light beam travelling toward them.
• b.A physicist moving at half the speed of light measures a light beam travelling toward them.
• c.A physicist moving at half the speed of light measures a light beam travelling away from them.

Following on from the police car scenarios, you might imagine that the relative speeds here would be 1.0, 1.5 and 0.5 times the speed of light. This would seem logical, but it’s not right! In all three cases the physicist would register the speed of the light beam as exactly the same value, measuring it at 3 × 10⁸ m/s. As this is a special value which is used all the time in science, it is generally written as just c.

Here’s Marcus again to discuss this development.

Download this video clip.Video player: Video 3

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Transcript: Video 3 Relative speed (part 2)

MARCUS DU SAUTOY

But what if the suspect car was moving at the speed of light? Well, now things get interesting! If the police are parked, and the suspect goes past at the speed of light, then, not surprisingly, 3x10⁸metres per second – the speed of light – usually referred to as c. But, if the suspect spaceship travels at the speed of light and the police are chasing at half the speed of light, then what would the speed gun show then?

Well, curiously, it wouldn’t be half the speed of light. It would be measured as c, exactly the same as if the police were stationary. And if they were going in the opposite direction? c again. Curious, eh? This was a problem that occupied physicists and astronomers in the late 19th century. They knew that the Earth was moving around the Sun at 30 kilometres per second. But no matter in what direction they measured the speed of light, it always came out the same. Well, how could this be?

The question was answered by Albert Einstein in 1905. In his theory of special relativity, he asserted that the speed of light in empty space is an unvarying, fundamental constant of nature. Everyone will measure the same value, no matter how fast they are moving. There is no ‘cosmic roadside’ where we have to stand to measure the speed of light properly.

On the face of it, this doesn’t make sense. But Einstein showed that the fixed speed of light was actually a consequence of something deeper: that ‘your time’ and ‘my time’ flow differently when we are in relative motion.

End transcript: Video 3 Relative speed (part 2)

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Video 3 Relative speed (part 2)

This will all begin to make sense again – but not before you reconsider the nature of time.