What does it mean to say that a car is moving at 80 km/h? Usually, it means

80 km/h relative to Earth. But Earth is moving at 30km/s relative to the sun,

so the car’s speed relative to the Sun is much greater. Statements about velocity

and speed are meaningful only when we have an answer to the question “velocity

relative to what?” The object or system with respect to which velocity is mea-

sured, Earth is our frame of reference. But playing tennis on the deck of a moving

ship, you would find the ship’s frame of reference more appropriate for describ-

ing the motion of the tennis ball. Similarly, the motion of a comet or interplan-

etary space probe is described most simply in the reference frame of the Sun.

We’ll designate different frames of reference using the symbols S and S’.

You’re speeding down the road in your sports car at 80 km/h. A police officer

behind you is moving at the legal speed limit of 50 km/h (see Fig.3-18). The

officer clocks your speed with radar. What does the radar indicate? Your posi-

tion is changing at the rate of 80 km every hour; the officer’s at the rate of

50 km/h. So the distance between you is increasing at 30 km/h. That’s your

relative speed－and that’s what the police radar reads. Since the police car has

a known speed of 50 km/h relative to the road, the officer can easily determine

that you’re going at

50 km/h + 30 km/h = 80 km/h

relative to the road. So you’ll get a speeding ticket！ We can put this more abstractly but also more generally. Suppose some

object (e.g., your car_ has velocity v (e.g.,80i km/h) relative to some frame of

reference designated S (e.g., the Earth). Suppose another reference frame S’

(e.g., the police car) is moving with velocity V( e.g.,50i km/h) relative to S.

Then the velocity v’ of the object relative to frame S’ is given by

v’ = v – V.

In our speeding example, this means that your speed relative to the police car

(i.e., your speed v’ relative to frame S’) is

v’ = 80i km/h - 50i km/h = 30i km/h.

We used the full vector formalism in this one-dimensional example because

Equation 3-10 holds regardless of the directions of motion of the object and the

reference frames. Example 3-6 illustrates the power of the relative velocity

Concept applied to a practical problem in navigation.