Mechanics · Quick Reference

Motion in Two Dimensions Cheat Sheet

Mechanics · Lesson 5/19

In one line: real-world motion rarely happens in a straight line — two-dimensional motion analyzes horizontal and vertical movement independently, then combines them.

Key Ideas

1Independence of Perpendicular Motion. Motion in the x-direction and y-direction can be analyzed completely separately, since forces or velocity in one direction don't affect the other.

2Position Vector. The position of an object in 2D can be written as (x,y), specifying its location along two perpendicular axes.

3Velocity Components. A velocity vector can be split into horizontal (vx) and vertical (vy) components, often using vx=v*cos(theta) and vy=v*sin(theta) for a velocity v at angle theta.

4Relative Velocity. The velocity of one object as observed from another moving object, found by vector subtraction: v_relative = v_A - v_B.

5Resultant Vector. The single combined vector representing the net effect of two or more vectors added together.

Worked Examples

A velocity has magnitude 10 m/s at angle 30° above horizontal. Find its horizontal component (vx). (cos(30°)~0.866)

vx ≈ 8.66 m/s

Using the same velocity, find its vertical component (vy). (sin(30°)=0.5)

vy = 5 m/s

A boat moves at 5 m/s east relative to water, and the water flows at 2 m/s east relative to the ground. Find the boat's velocity relative to the ground.

7 m/s east

Formulas

vx = v*cos(theta)

vy = v*sin(theta)

Resultant magnitude = sqrt(vx^2+vy^2)

Practice Yourself

A velocity has magnitude 20 m/s at 0° (purely horizontal). Find vx and vy.

vx=20, vy=0

A velocity has magnitude 10 m/s at 90° (purely vertical). Find vx and vy.

vx=0, vy=10

What does 'independence of perpendicular motion' mean?

Horizontal and vertical motion can be analyzed completely separately