Definition: 2D vectors

2D vectors have vectors have magnitude and direction, but unlike 1D vectors, their direction is not just plus and minus -- we have to deal with two possible components. Good intuition for them is arrows. You should be comfortable with adding vectors:

• Think of them as "arrows"
• You add vectors tip to tail
• You can also break them up into components and add them component-wise.

Sometimes you might have to use trig functions to find the components -- a good tip is to remember sine is small when $\theta$ is small!

Idea: 2-D Reference frames

When working with reference frames in 2 dimensions, you must perform vector addition. For example, if you throw a ball at velocity $\vec v=u$ in the frame of a car moving at velocity $\vec v$, it moves at velocity $\vec v + \vec u$ in the frame of the ground.

Theorem: Projectile Motion

When a projectile moves through multiple dimensions, its motion in those dimensions can be treated independently.

• y-directional motion often determines how long an object is in the air for
• A projectile experiences no gravitational acceleration in the x-direction
• Combine information in both dimensions to derive the trajectory of the projectile, which is always a parabola

Tip: Helpful 2-D Kinematic equations

The range is given by $$R = \frac{v^2 \sin (2\theta)}{g}.$$ The total time in the air is given by $$t = \frac{2v\sin \theta}{g}.$$ The mximum height reached is given by $$h = \frac{v^2 \sin^2 \theta}{g}.$$