Reference frames and relative velocity
Every velocity you state is measured against some frame, and switching frames is simple vector subtraction.
Reference frames
A frame is the viewpoint you measure from, often the ground. The same object has different velocities in different frames: to a passenger on a smooth train, a dropped ball falls straight down, but to someone on the ground it traces a curve. Position, velocity, and acceleration all depend on the chosen frame.
\vec{v}_{A/\text{ground}} = \vec{v}_{A/B} + \vec{v}_{B/\text{ground}}Relative velocity vectors
The velocity of A as seen from B is the vector difference of their ground velocities. If two cars both head east at 25 m/s, each sees the other as stationary. If one heads east at 20 m/s and the other west at 15 m/s, each sees the other approaching at 35 m/s. Because it is a vector equation, directions must be handled with signs or components, never raw speeds alone.
\vec{v}_{A/B} = \vec{v}_{A/G} - \vec{v}_{B/G}Worked example
A boat points straight across a river at 4 m/s relative to the water while the current flows downstream at 3 m/s, and the river is 80 m wide. The crossing time depends only on the across-component: 80/4 = 20 s. The current does not speed up or slow down the crossing; it just carries the boat 3 times 20 = 60 m downstream, so the boat actually moves at 5 m/s along a diagonal at arctan(3/4) = 37 degrees from straight across.
|\vec{v}| = \sqrt{4^2 + 3^2} = 5 \text{ m/s}, \quad t = \frac{80}{4} = 20 \text{ s}Flashcard
Velocity of A relative to B
Check your understanding
A swimmer moves at 2.0 m/s straight across a 60 m wide river while a 1.5 m/s current runs parallel to the banks. How long does the crossing take?