Motion in a Plane

Topic-by-topic explanations

1. Introduction

Motion in a plane requires both magnitude and direction. Resolve vector quantities into perpendicular components and treat each component as a one-dimensional problem.

2. Scalars and vectors

Scalars have magnitude only (mass, time, temperature); vectors have magnitude and direction (displacement, velocity, acceleration, force). Vector equality requires both magnitude and direction to match.

3. Multiplication of vectors by real numbers

Multiplying A by a positive scalar changes its magnitude; multiplying by a negative scalar also reverses its direction. Multiplying by zero gives the zero vector.

4. Addition and subtraction of vectors – graphical method

Graphically add vectors head-to-tail or with the parallelogram rule. To subtract B, add −B. The resultant points from the first tail to the final head.

5. Resolution of vectors

For a vector A at angle θ from +x, Aₓ=A cosθ and Aᵧ=A sinθ. Check the angle convention and signs when the vector is in another quadrant.

6. Vector addition – analytical method

Add components: Rₓ=ΣAₓ and Rᵧ=ΣAᵧ. Then R=√(Rₓ²+Rᵧ²); determine the direction using both components so the correct quadrant is selected.

7. Motion in a plane

Position, velocity and acceleration in a plane each have x and y components. The component equations share the same time interval and can be recombined into the resultant vector.

8. Motion in a plane with constant acceleration

For constant acceleration in a plane, use sₓ=uₓt+½aₓt² and sᵧ=uᵧt+½aᵧt², with separate velocity equations. Gravity acts vertically downward when air resistance is neglected.

9. Projectile motion

In ideal projectile motion, horizontal acceleration is zero and vertical acceleration is −g. For equal launch and landing levels, T=2u sinθ/g, H=u²sin²θ/(2g), and R=u²sin2θ/g.

10. Uniform circular motion

Uniform circular motion has constant speed but changing velocity direction. The inward acceleration is a_c=v²/r=ω²r, and the relations v=rω and T=2π/ω connect linear and angular motion.

Formula sheet

Use consistent SI units, identify the system and assumptions, and show the steps of each calculation. Check units and whether the result is physically reasonable.

Worked example

Common errors and revision checklist

Do not add vector magnitudes directly unless directions allow it. Keep axes consistent, and remember centripetal acceleration points inward.

• Write the relevant law or relation before substituting values.
• Show units, signs and directions.
• Check assumptions and round only the final result.