Changing Frames

Position, velocity and acceleration all depend on who is measuring. This lesson gives the rules for translating between observers, and explains why you tilt your umbrella when you run through rain.

Positions in two frames#

Let frames S and S′ watch the same particle P. The frames may be moving relative to each other. With O and O′ their origins, the position of P relative to S is $\vec r_{P,S} = \vec{OP}$ and relative to S′ is $\vec r_{P,S'} = \vec{O'P}$, while the position of S′ itself (its origin) relative to S is $\vec r_{S',S} = \vec{OO'}$. The triangle rule links them:

Velocities in two frames#

Differentiate (3.26) with respect to time:

The velocity of the particle relative to S is its velocity relative to S′ plus the velocity of S′ relative to S. Rearranged, it answers the everyday question “how fast is A moving as seen from B?”:

Two everyday cases. A rifle fires bullets at 80 m/s relative to the rifle. Mounted on a train moving at 25 m/s and fired forward, the bullet moves at 80 + 25 = 105 m/s relative to the ground. And a swimmer who manages 1.5 m/s in still water (that is, relative to the water) swimming with a 1.0 m/s current moves at 2.5 m/s relative to the bank.

Accelerations in two frames#

Differentiate once more:

If S′ moves relative to S at constant velocity, then $\vec a_{S',S} = 0$ and $\vec a_{P,S} = \vec a_{P,S'}$. Observers drifting past each other uniformly disagree about positions and velocities, but they agree about accelerations. That fact will become the foundation of Newton’s laws in the chapters ahead.