Short on time? This page is the recap. For explanations, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A reference frame is an origin, an axis and a clock that belong to one observer. Every position and velocity is measured in some frame.
- Changing the observer can change the size and direction of a measured velocity or displacement.
- An inertial frame is not accelerating. Treat every frame as inertial unless told otherwise.
- In this course, relative velocities are combined along one line only, as signed numbers.
- All inertial observers measure the same acceleration and the same time interval Δt.
Key relationships
| Relationship | Meaning | Use it to |
|---|---|---|
| v_AC = v_AB + v_BC | inner letters match | find what an outside observer measures |
| v_BA = −v_AB | swap subscripts, flip sign | describe the observer as seen by the object |
| v_AB = v_AG − v_BG | object first, observer second | convert two ground-frame velocities |
| x_AB = x_AG − x_BG | position of A measured by B | convert positions |
| Δx_AC = Δx_AB + Δx_BC | displacements chain too | compare how far each observer says A moved |
| a_AB = a_AG (B inertial) | acceleration is frame-independent | skip recalculating acceleration |
Assumptions behind the numbers
- Objects and observers are points moving on one straight line.
- Every frame moves at constant velocity relative to the ground, and the ground counts as inertial.
- All observers’ clocks agree. (This fails only near the speed of light, which is outside the course.)
- Near Earth’s surface, g = 9.8 m/s² downward, the same in every inertial frame.
Mistakes to avoid
- No observer named. “The velocity is 5 m/s” relative to what?
- Subtracting speeds instead of signed velocities. Objects approaching each other close at the sum of their speeds.
- Subscripts reversed. v_AG − v_BG gives v_AB, not v_BA.
- Recalculating acceleration in a moving frame. For an inertial observer it is unchanged.
- Thinking one observer is wrong. Both are right in their own frames.
- Adding velocities at an angle. Relative motion here stays on one line.
Quick self-check
- Take +x downstream. A river flows at +1.0 m/s. A boat moves at 3.0 m/s relative to the water. What is its velocity relative to the bank going downstream, and going upstream? (+4.0 m/s; −2.0 m/s)
- v_AB = +6.0 m/s. What is v_BA? (−6.0 m/s)
- A car accelerates at 2.5 m/s² relative to the road. What acceleration does a bus driver moving at a steady 12 m/s measure for it? (2.5 m/s², the same)
- With +x east, an object is at x = 40 m and an observer at x = 55 m, both measured from the ground. Where is the object in the observer’s frame? (−15 m: 15 m west of the observer)
Next: practice questions.