Short on time? This page is the recap. For explanations, graphs and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Treat the moving thing as a point object. Choose an origin and a positive direction, and write them down.
- Position x says where the object is. Displacement Δx is the change in position. Distance travelled is the total path length and is never negative.
- Average velocity uses only the start and end of an interval. Average speed uses the whole path.
- An object accelerates whenever its velocity changes: faster, slower, or reversing direction.
- Instantaneous velocity (or acceleration) is the average over a very short interval. On a graph, it is the slope of the tangent line.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| Δx = x − x₀ | x, x₀ in m | find displacement (signed) |
| v_avg = Δx / Δt | m/s | find average velocity over an interval |
| average speed = distance / Δt | m/s, never negative | describe the whole path, including turn-arounds |
| a_avg = Δv_x / Δt = (v_x − v_x0) / Δt | m/s² | find average acceleration |
| slope of x–t graph | m/s | velocity (tangent slope = instantaneous) |
| slope of v_x–t graph | m/s² | acceleration |
| area under v_x–t graph | m | displacement (area below the axis is negative) |
Assumptions behind the numbers
- The object is a point: rotation and size are ignored.
- Motion is along one straight line, so a sign fully describes direction.
- The reference frame is at rest relative to the ground (inertial) unless stated.
- Near Earth’s surface, free-fall acceleration has size g = 9.8 m/s² downward (air resistance ignored). The course also accepts 10 m/s² where stated.
Mistakes to avoid
- No axis stated. A sign means nothing without one.
- Negative a = slowing down. Wrong. Compare the signs of v_x and a_x.
- Dropping signs in Δv_x. A ball that rebounds changes velocity by more than the change in speed.
- Mean of two velocities as average velocity. Use total Δx ÷ total Δt.
- Secant slope used as the instantaneous value. Use a tangent, or shrink the interval.
- v = 0 so a = 0. Not at the top of a throw.
- Distance and displacement swapped. Check whether the object turned around.
Quick self-check
- With +x to the right, a cart goes from x = +2.0 m to x = −4.0 m in 3.0 s. What is its average velocity? (−2.0 m/s)
- v_x = −5.0 m/s and a_x = −2.0 m/s². Speeding up or slowing down? (Speeding up: same signs.)
- A ball hits a wall at +3.0 m/s and rebounds at −2.0 m/s after 0.10 s of contact. What is its average acceleration? (−50 m/s²)
Next: practice questions.