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
- Any vector can be replaced by two perpendicular components along axes you choose. Components are signed; the magnitude is never negative.
- You may pick any perpendicular axes. Choose the ones that make the problem simplest, such as axes along a ramp.
- To add vectors, add the x-components, add the y-components, then rebuild the resultant.
- Motion in a plane is two one-dimensional motions. Each axis uses the Topic 1.3 equations; the time t is shared.
- A projectile has zero horizontal acceleration and constant downward vertical acceleration g.
Key relationships
| Relationship | Notes | Use it to |
|---|---|---|
| A_x = A cos θ, A_y = A sin θ | θ measured from the +x axis | resolve a vector |
| A = √(A_x² + A_y²) | Pythagoras | find the magnitude |
| tan θ = A_y / A_x | check the quadrant with a sketch | find the direction |
| R_x = A_x + B_x, R_y = A_y + B_y | signed components | add vectors |
| v_x0 = v₀ cos θ, v_y0 = v₀ sin θ | θ above the horizontal | start a projectile problem |
| x = x₀ + v_x0 t | a_x = 0 | horizontal motion |
| v_y = v_y0 − g t, y = y₀ + v_y0 t − ½ g t² | +y up | vertical motion |
| t = √(2h / g) | launched horizontally from height h | fall time |
| R = 2v_x0 v_y0 / g | launch and landing at the same level | range on level ground |
Assumptions behind the numbers
- The object is a point; air resistance is negligible unless stated.
- The reference frame is inertial (not accelerating; the ground counts as one) unless stated.
- Near Earth’s surface, g = 9.8 m/s² downward and constant.
- Combining velocities between moving frames stays in one dimension in this course (Topic 1.4).
Mistakes to avoid
- Using cos for x every time. Only when θ is measured from the x-axis.
- Trusting tan⁻¹ blindly. It returns −90° to +90°. Sketch to pick the quadrant.
- Adding magnitudes. Add components instead.
- Zero velocity at the top. Only v_y is zero; the speed there is v_x0.
- Zero acceleration at the top. It is g downward throughout.
- Letting v_x change in flight. With no air resistance, a_x = 0.
- Mixing axes in one equation. Keep x-quantities and y-quantities apart; only t crosses over.
Quick self-check
- Resolve 10 m/s at 30° above the +x axis. (v_x = 8.7 m/s, v_y = 5.0 m/s)
- A displacement has components (−6.0 m, +8.0 m). Find its size and direction. (10 m, 53° above the −x axis, i.e. 127° from +x)
- A ball rolls off a 0.80 m high table at 2.0 m/s. How long is it in the air, and how far from the table does it land? (0.40 s; 0.81 m)
Next: practice questions.