Unit 1.5: Vectors and Motion in Two Dimensions

Introduction

In this section, we extend our understanding of vectors into two dimensions. Many real-world motion problems, such as projectile motion and river crossings, require analyzing motion along two perpendicular axes—typically horizontal (\(x\)) and vertical (\(y\)).

The ability to resolve vectors into components and use vector addition is essential for solving problems involving two-dimensional motion.

Key Concepts

Vector Components: Any vector can be broken into horizontal (\(x\)) and vertical (\(y\)) components using trigonometry:
- \( V_x = V \cos \theta \)
- \( V_y = V \sin \theta \)
Vector Addition: When adding vectors that are not collinear, break them into components first, then sum components separately:
- \( R_x = A_x + B_x \)
- \( R_y = A_y + B_y \)
The magnitude of the resultant vector is then given by:
\( R = \sqrt{R_x^2 + R_y^2} \)
and its direction by:
\( \theta = \tan^{-1} \left(\frac{R_y}{R_x}\right) \)
Projectile Motion: Motion in two dimensions where the only force acting is gravity. The horizontal and vertical motions are independent of each other.

Tip: In projectile motion, horizontal velocity remains constant, while vertical velocity changes due to gravity.

Graphical Representations

Vector problems often involve graphical methods of solution:

Use tip-to-tail method for adding vectors visually.
Draw right triangles to break vectors into components.
Sketching motion diagrams helps in analyzing projectile motion trajectories.

Exam Strategy: If solving a vector problem, sketch a diagram first to clarify relationships between components.

Mathematical Routines

Solving two-dimensional motion problems requires careful application of kinematics and trigonometry:

Identify knowns and unknowns in each dimension separately.
Apply kinematic equations separately in horizontal and vertical directions.
Use Pythagorean theorem and inverse tangent to find resultant displacement or velocity.

Tip: Always resolve vectors into components first before applying kinematic equations.

Practice Activities

Activity 1: Resolving a Vector

A boat moves at 5 m/s in a direction 30° north of east. Determine the boat’s eastward and northward velocity components.

Activity 2: Adding Two Vectors

A plane is flying 100 m/s east, and the wind is blowing 40 m/s north. Determine the resultant velocity of the plane.

Activity 3: Projectile Motion

A soccer ball is kicked at 20 m/s at a 45° angle. Determine the ball’s initial horizontal and vertical velocity components.

Summary & Exam Preparation Tips

Understanding vectors and motion in two dimensions is essential for solving physics problems involving projectile motion and relative velocity. Key takeaways include:

Always resolve vectors into horizontal and vertical components.
Use the Pythagorean theorem to find resultant magnitude and inverse tangent for direction.
Projectile motion follows two separate 1D kinematics equations.
Velocity components remain independent—gravity only affects vertical motion.

Practicing vector addition and projectile motion problems will improve problem-solving skills and help with AP Physics free-response and multiple-choice questions.