AP Physics C/Vectors and Motion in Two Dimensions

Vectors in 2D

Vectors in 2 dimensions have both an x and a y component. We can find the components if given an angle, θ, in the following manner:

$v_{0x}=v_{0}\cos {\theta }$

$v_{0y}=v_{0}\sin {\theta }$

Projectile motion

Projectile motion is one of the most common applications of 2D vectors. Lets start with an example, lets say we throw a ball 30 m/s at a 60° angle. We can figure out how far it will land and how long it will take. The acceleration in the y of a projectile motion question is always $a_{y}=-g=-9.8m/s$. Since velocity is the integral of acceleration, $v_{y}=\int a_{y}dt=a_{y}t+C$, and C will always be equal to initial velocity, $v_{y}=v_{0y}+a_{y}t\implies v_{y}=v_{0}\sin {\theta }+a_{y}t$ .

IMPORTANT: Any projectile with an initial height or y of 0 will reach the apex of its height at $t_{0.5}$ (where t is the total time of travel). This is because at the apex, the slope of the tangent line to the y position is equal to zero.

So lets solve for time. $v_{y}=v_{0}\sin {\theta }+a_{y}t\implies 0=30\sin {60}-9.8t\implies -30\sin {60}/-9.8=t\implies t=2.65s$

From here we can solve for the range of the object.

$a_{x}=0$

$v_{x}=v\cos {\theta }$

$x=\int v_{x}\implies \int v\cos {\theta }$

$x=v\cos {\theta }*t$

$x=30\cos {60}*2.65\implies x=39.8m$