Short on time? This page is the recap. For diagrams, derivations and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- One motion can be shown as words, a motion diagram, graphs, equations or a data table. They must all agree.
- Motion diagram: dots at equal time intervals. Growing gaps mean speeding up; shrinking gaps mean slowing down. The acceleration arrow shows which way the velocity is changing.
- Graphs are linked by slopes (tangent slopes for curves) and areas.
- With constant acceleration, the v_x–t graph is a straight line and three equations apply.
- Free fall: near Earth’s surface, a_y is constant and downward with size g ≈ 9.8 m/s².
- Changing acceleration: describe it, sketch it, but do not calculate with the constant-acceleration equations.
Key relationships
| Relationship | Meaning | Note |
|---|---|---|
| slope of x–t | v_x (m/s) | tangent slope gives the instantaneous value |
| slope of v_x–t | a_x (m/s²) | a falling line means a_x < 0, not moving backwards |
| area under v_x–t | Δx (m) | area below the axis is negative |
| area under a_x–t | Δv_x (m/s) | add v_x0 to get the final velocity |
| v_x = v_x0 + a_x t | leaves out x | constant a_x only |
| x = x₀ + v_x0 t + ½a_x t² | leaves out v_x | constant a_x only |
| v_x² = v_x0² + 2a_x(x − x₀) | leaves out t | choose the root from the direction |
| a_y = −g (with +y up) | free fall | same on the way up, at the top and on the way down |
Assumptions
- The object is a point moving along one line.
- Acceleration is constant within each stage you calculate.
- In free fall, air resistance is ignored and the object is near Earth’s surface.
Mistakes to avoid
- Reading an x–t graph as a picture of the path.
- Using the kinematic equations when a_x changes.
- Picking the wrong root of v_x²; check the direction of motion.
- Writing a_y = 0 at the top of a throw.
- Forgetting that area below the time axis is negative.
- Using a_y = +g after choosing +y upward.
- Treating the a_x–t area as the final velocity instead of the change in velocity.
Quick self-check
- A cart starts at +2.0 m/s and accelerates at +1.5 m/s² for 4.0 s. Find its velocity and displacement. (+8.0 m/s; +20 m)
- A stone is dropped from rest. How far does it fall in 1.5 s? Use g = 9.8 m/s². (11 m)
- An object moving at +5.0 m/s has a_x = −2.0 m/s² for 3.0 s. Find its final velocity. (−1.0 m/s; it has turned around)
Next: practice questions.