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, $v_y=v_{0y}+a_{y}t⟹v_y=v_0\sin \theta +a_{y}t$ .

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. $v_y=v_0\sin \theta +a_{y}t⟹0=30\sin 60-9.8t⟹-30\sin 60/-9.8=t⟹t=2.65s$

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

$a_x=0$

$v_x=v\cos \theta$

$x=\int v_x⟹\int v\cos \theta$

$x=v\cos \theta *t$

$x=30\cos 60*2.65⟹x=39.8m$