Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For diagrams and worked examples, use the full study guide.
Recap
- One motion can be shown as words, a figure, a motion diagram, graphs or equations. Be ready to turn any one into any other.
- State the axis first, for example “+y upward, origin at the launch point”.
- Motion diagram: dots at equal time steps. Growing gaps mean speeding up; shrinking gaps mean slowing down. Equal changes in the gaps mean constant acceleration.
- At a turning point, v = 0 but the acceleration need not be zero.
Key relationships
| Relationship | Graph meaning |
|---|---|
| v_x = dx/dt | slope of tangent to x–t |
| a_x = dv_x/dt = d²x/dt² | slope of tangent to v_x–t; curvature of x–t |
| Δx = ∫ v_x dt | signed area under v_x–t |
| Δv_x = ∫ a_x dt | signed area under a_x–t |
| v_x = v_x0 + a_x t | no x; constant a_x only |
| x = x₀ + v_x0 t + ½a_x t² | no v_x; constant a_x only |
| v_x² = v_x0² + 2a_x(x − x₀) | no t; constant a_x only |
Assumptions behind the numbers
- Point object, one straight line, one stage of constant acceleration at a time.
- Free fall near Earth: a = g ≈ 9.8 m/s², downward and constant; no air resistance. The course accepts g = 10 m/s², 9.8 m/s² or 9.81 m/s². State your choice.
- Factors of change: write the result as a proportion first, for example h = v₀²/(2g), so h ∝ v₀².
Mistakes to avoid
- Reading the height of x–t as the velocity. Velocity is the slope.
- Area under a_x–t = v_x. It is Δv_x; add v_x0.
- a = 0 at the top of a throw. It stays −g.
- One equation across stages. Restart at each change of acceleration.
- Corners on x–t. Velocity cannot jump, so x–t bends smoothly.
- Double negatives with g. With +y up, a_y = −9.8 m/s², not −(−9.8).
- Keeping both quadratic roots. Reject the root that is before the motion starts.
Quick self-check
- A ball is thrown straight up at 7.0 m/s. How high does it rise? (h = v₀²/(2g) = 49 ÷ 19.6 = 2.5 m)
- An a_x–t graph is flat at 1.5 m/s² for 4.0 s, and v_x0 = −2.0 m/s. What is v_x at the end? (Area 6.0 m/s, so v_x = −2.0 + 6.0 = 4.0 m/s)
- A stone dropped from rest falls a height h in time T. How long does it take to fall 4h? (t ∝ √h, so 2T)
Next: practice questions.