Motion in Two Dimensions Finding the net direction of two motions Use Pythagorean Theorem to find the magnitude of the resultant vector Use the tangent function to find the direction (angle) of resultant motion tan θ = opp/adj θ = tan-1(opp/adj)

Motion in Two Dimensions Resolving a vector into components The horizontal component is the vector (here = F) times cos θ The vertical component is the vector (again = F) time sin θ

Projectile Motion Launched horizontally (cliff top) Horizontal motion is uniform motion (no acceleration) vx,I = vx,t = vx,f = constant Vertical motion is uniform accelerated motion Use equations for falling objects!! Link H + V motion with time

Projectile Motion Launched at an angle (full parabola) Initial velocity is at an angle (the resultant vector) Find the horizontal and vertical components of the initial velocity Horizontal motion is still at constant velocity! Vertical motion is still at constant acceleration! Continue to use time to link horizontal and vertical motion!

Uniform Circular Motion Recall that velocity is described by magnitude (speed) and direction. Sometimes the acceleration an object experiences changes its direction and not its speed. This occurs in uniform circular motion – circular motion at a constant (tangential) velocity Objects in uniform circular motion experience “centripetal” acceleration This acceleration is directed from the object into the center of the circle

Centripetal Acceleration Centripetal acceleration is Proportional to the square of the tangential velocity Inversely proportional to the radius of the circle ac = v2 / r

Centripetal Force If Force = mass x acceleration… Centripetal force = mass x centripetal acceleration Fc = m v2 / r Centripetal force is also directed toward the center of circular motion

Implications If the centripetal acceleration/force is removed, the object will move in the direction of its tangential velocity Centripetal force (due to the sun’s gravity) keeps Earth in its orbit