## Velocity changes position

Physics Narrative for 14-16

#### Velocity and changing position

Alice, Bob and Charlie are once again at the railway station (no rest for good behaviour). Charlie is on the platform and Alice is in the up-train whilst Bob is in the down-train.

The up-train moves past: Charlie notices that it gets 14 metre farther away from him each second. He might have written this as velocity of 14 metre inverse second. Next, he notices Bob on the down-train: Charlie notices that the separation between Bob and himself decreases by 8 metre inverse second. The direction is reversed, and to bring attention to this feature he records the velocity as -8 metre inverse second.

How will Alice and Bob record Charlie's velocity?

And what about Charlie_{displaced}, on another platform just along the way?

Surely he'll see the up-train gets 14 metre closer to him each second, so he'll record the velocity as -14 metre inverse second. But the down-train is also moving so that the separation between Bob and himself decreases by 8 metre inverse second. So he records the velocity as -8 metre inverse second.

Charlie and Charlie_{displaced} are consistent in taking velocity as change in positiontime, and yet they disagree about the velocities, even through they're recording values for the same trains, at the same time. There is something strange going on here, and we'd better pay attention to thinking about how we introduce the relationship between change in position, or displacement, and velocity.

#### Understanding velocity – it's essentially a vector

Treated as vectors, there's no problem.

The →v always determines the change in →s (the →d) in each second. There's an elegant simplicity here, if you use the most helpful formalism. (The difficulties in establishing consistency between Charlie and Charlie_{displaced} came with trying to simplify the vectors down to a single dimension and treat them as a signed quantity.)

displacement = velocity × duration (Or, in symbols, Δ *d* = *v* × Δ *t*).

Velocity has the same relationship to displacement (so to change in position) as acceleration does to velocity. The common pattern is one to keep in mind, as then everything you learn about the behaviour of one pair can easily be transferred to the second pair.