Motion in a Plane

The real world isn't one-dimensional. A soccer ball flies through the air, an airplane banks and turns, a planet orbits in space—all in two dimensions or more. When motion is confined to a plane (two dimensions), we must think in terms of vectors: quantities with both magnitude and direction. This chapter builds vector mathematics from scratch and applies it to describe projectile motion, circular motion, and relative motion in two dimensions.

Scalars vs. Vectors

Scalars are quantities with only magnitude: temperature (25°C), mass (5 kg), speed (10 m/s). You describe them with a single number.

Vectors have both magnitude and direction: displacement (5 meters north), velocity (10 m/s northeast), force (50 newtons downward). They need direction to be fully specified.

Think of scalars as quantities that "have size" and vectors as quantities that "point somewhere." When combining scalars, you simply add their numbers. With vectors, direction matters crucially.

Vector Representation

Vectors are drawn as arrows: the length represents magnitude, the arrowhead shows direction. In mathematical notation, we use bold letters (v) or letters with an arrow (⃗v) to distinguish vectors from scalars.

A vector in a plane can be decomposed into components: horizontal (x) and vertical (y) parts. A vector of magnitude 10 m/s at 37° from horizontal has:

• x-component: 10 cos(37°) ≈ 8 m/s
• y-component: 10 sin(37°) ≈ 6 m/s

Components are powerful because they let you handle 2D problems as two separate 1D problems. Motion in the x-direction is independent of motion in the y-direction (in most cases).

Vector Operations

Vector Addition: To add two vectors, place the tail of the second vector at the head of the first. The sum vector connects the starting point to the final point. Algebraically, add components: (A_x + B_x, A_y + B_y).

Vector Subtraction: Subtract by adding the negative of a vector. A − B = A + (−B).

Multiplication by a Scalar: Multiplying a vector by a positive number stretches or shrinks it; multiplying by a negative number reverses its direction.

Dot Product (Scalar Product): A · B = |A||B| cos(θ), where θ is the angle between them. This produces a scalar. For example, work = force · displacement.

Cross Product (Vector Product): A × B produces a vector perpendicular to both A and B, with magnitude |A||B| sin(θ). For example, torque = position × force.

Projectile Motion: Nature's Parabola

Drop a ball, and it falls straight down. Throw it horizontally, and it follows a curved path. Throw it at an angle, and it traces an arc. These are all projectile motion—motion under constant gravitational acceleration.

The magic of projectile motion is that horizontal and vertical motions are independent. Gravity acts only vertically, so:

• Horizontal: constant velocity (no acceleration)
• Vertical: constant acceleration (g ≈ 10 m/s² downward)

Treat them separately! If you throw a ball at 20 m/s horizontally and it takes 2 seconds to hit the ground:

• Horizontal distance: 20 m/s × 2 s = 40 meters
• Vertical distance: ½ × 10 × 2² = 20 meters

For a projectile launched at angle θ with initial speed u:

• Maximum height: occurs when vertical velocity becomes zero
• Range: maximum horizontal distance, maximized at 45°
• Time of flight: determined by vertical motion alone

The trajectory is a parabola. At the peak, vertical velocity is zero, but horizontal velocity remains constant. This is why a projectile "hangs" briefly at the top—the vertical velocity component vanishes.

Circular Motion

When an object moves in a circle at constant speed, is it accelerating? Yes! Its direction constantly changes, so its velocity vector constantly changes. This centripetal acceleration points toward the center of the circle.

For circular motion at speed v in a circle of radius r:

• Centripetal acceleration: a = v²/r
• This acceleration requires a centripetal force (toward center) of F = mv²/r

Think of a ball on a string: tension provides the centripetal force. On a banked curve, gravity and normal force combine to provide centripetal force. Without this inward force, the object flies off tangentially.

Relative Velocity in 2D

If a boat moves at 10 m/s north and the current is 3 m/s east, the boat's velocity relative to the shore is the vector sum: √(10² + 3²) ≈ 10.4 m/s at an angle. This extends straight-line relative velocity to two dimensions.

The principle remains: add velocity vectors to find relative velocities. The boat's velocity relative to water plus water's velocity relative to shore equals boat's velocity relative to shore.

Socratic Questions

1. Why is it that you can treat horizontal and vertical motion independently in projectile motion? What would be different if air resistance acted sideways instead of just downward?

1. A baseball pitcher throws a fastball horizontally at 40 m/s. By the time it reaches home plate (18 meters away), gravity has pulled it down. How far down has it fallen? (Hint: How long does the ball take to travel 18 meters horizontally?)

1. In circular motion at constant speed, why do we say the object is accelerating? What direction does this acceleration point, and what force provides it?

1. If you're in a moving car and throw a ball forward, why does the ball's path relative to the ground look different from the path relative to someone inside the car? How do velocity vectors explain this?

1. At what launch angle does a projectile achieve maximum range? Why is this angle the same regardless of the launch speed?