Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which limits relative velocity to one dimension). For explanations, diagrams and worked examples, use the full study guide.
Recap
- A reference frame is an observer’s origin, axes, clock and motion. State it before any number, for example “+x east, measured from the ground”.
- The frame decides both the size and the direction of a measured position or velocity.
- Subscripts: v_PA means the velocity of P as measured in frame A.
- Converting: add or subtract vectors, not speeds. Use components when the vectors are not parallel.
- Inertial frames do not accelerate. Assume a frame is inertial unless told otherwise.
Key relationships
| Relationship | Meaning | Condition |
|---|---|---|
| r_PA = r_PB + r_BA | positions link through frame B | always |
| v_PA = v_PB + v_BA | inner letters match and drop out | everyday speeds |
| v_BA = −v_AB | swapping the order flips the vector | always |
| x′ = x − Vt, v′ = v − V | one-dimensional frame change | S′ moves at constant V along +x; origins match at t = 0 |
| a_PA = a_PB + a_BA | derivative of the velocity link | always |
| a_PA = a_PB | same acceleration in both frames | a_BA = 0 (both inertial) |
| Track straight across a crosswind w at airspeed u | point into the wind at θ from the track, sin θ = w/u; ground speed √(u² − w²) | w < u |
Assumptions behind the numbers
- Frames are inertial (constant velocity relative to the ground) unless stated.
- Speeds are far below the speed of light, so velocities add as vectors.
- Free fall: a_y = −g with g = 9.8 m/s² (+y upward), air resistance ignored.
Mistakes to avoid
- Adding speeds as numbers when the velocities are not parallel. Use components.
- Wrong subscript order. Write the chain so the inner letters match.
- Dropping a sign. v_BA = −v_AB, so the minus sign carries direction.
- Pointing = travelling. The heading (relative to air or water) is not the track over the ground.
- Different accelerations in different inertial frames. They are the same.
- Forgetting the initial velocity an object shares with the frame it leaves.
Quick self-check
- A boat moves at 3.0 m/s relative to the water in a river flowing at 1.0 m/s. What is its speed relative to the bank going downstream, and upstream? (4.0 m/s and 2.0 m/s)
- A plane flies north at 80 m/s relative to the air; the wind blows east at 60 m/s. What is its ground velocity? (100 m/s at 36.9° east of north)
- v_AB = +4.0 m/s along +x. What is v_BA? (−4.0 m/s)
Next: practice questions.