Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which uses no calculus). For explanations, diagrams and worked examples, use the full study guide.
Recap
- State the axes first, for example “+x horizontal, +y up”. Write position as r(t) = x(t) i + y(t) j.
- Motion in a plane is two one-dimensional motions. Fill an x | y table: initial position, initial velocity, acceleration.
- The components can have different velocities and accelerations, and either can change with time.
- Changing the motion along one axis does not change the motion along the perpendicular axis.
- A projectile has zero acceleration horizontally and a constant acceleration g downward.
- Three dimensions add a z-component in the same way (background only). This course analyses motion in two dimensions.
Key relationships
| Relationship | Notes |
|---|---|
| v_x = dx/dt, v_y = dy/dt | velocity is tangent to the path |
| a_x = dv_x/dt, a_y = dv_y/dt | a need not be parallel to v |
| v_x = v_x0 + ∫a_x dt, x = x₀ + ∫v_x dt (same for y) | one initial condition per component |
| |v| = √(v_x² + v_y²), tan θ = v_y / v_x | check the quadrant with a sketch |
| Projectile: x = v₀ cos θ · t, y = v₀ sin θ · t − ½gt² | +y up, launch from origin |
| y = x tan θ − gx² / (2v₀² cos²θ) | path is a parabola |
| R = v₀² sin 2θ / g | equal launch and landing heights only |
| Horizontal launch from height h: t = √(2h/g) | fall time does not depend on horizontal speed |
Assumptions behind the numbers
- Point object; inertial frame (the ground) unless stated.
- Projectiles: air resistance ignored, g = 9.8 m/s² downward and constant.
- Coefficients in r(t) carry units (in x = 0.10t³, the 0.10 is in m/s³).
Mistakes to avoid
- Speed zero at the top. Only v_y = 0; the speed is v_x0.
- Range formula on a cliff or slope. Solve y(t) = landing height instead.
- Constant-a equations for a component that changes. Integrate it.
- Adding component sizes (3 and 4 make 5, not 7).
- Different times for x and y. Find t from one component and use it in the other.
- Treating v and a as parallel. For a projectile, a always points down.
Quick self-check
- r(t) = (2.0t²) i + (3.0t) j (m). What is the speed at t = 1.0 s? (v = 4.0 i + 3.0 j, so 5.0 m/s)
- A ball rolls off a 1.25 m high table at 2.0 m/s. How far from the table does it land? (t = √(2 × 1.25 ÷ 9.8) = 0.51 s; x = 1.0 m)
- A ball is launched at 20 m/s, 60° above the horizontal. What are its speed and acceleration at the top? (10 m/s; 9.8 m/s² downward)
Next: practice questions.