R. Field 1/22/2013 University of Florida PHY 2053Page 1 2-d Motion: Constant Acceleration Kinematic Equations of Motion (Vector Form) The velocity vector and position vector are a function of the time t. Acceleration Vector (constant) Velocity Vector (function of t) Position Vector (function of t) Velocity Vector at time t = 0. Position Vector at time t = 0. The components of the acceleration vector, a x and a y, are constants. The components of the velocity vector at t = 0, v x0 and v y0, are constants. The components of the position vector at t = 0, x 0 and y 0, are constants. Warning! These equations are only valid if the acceleration is constant.

R. Field 1/22/2013 University of Florida PHY 2053Page 2 2-d Motion: Constant Acceleration Kinematic Equations of Motion (Component Form) constant The components of the acceleration vector, a x and a y, are constants. The components of the velocity vector at t = 0, v x0 and v y0, are constants. The components of the position vector at t = 0, x 0 and y 0, are constants. Warning! These equations are only valid if the acceleration is constant. constant Ancillary Equations Valid at any time t

R. Field 1/22/2013 University of Florida PHY 2053Page 3 Acceleration Due to Gravity Experimental Result Earth y-axis x-axis RERE h Near the surface of the Earth all objects fall toward the center of the Earth with the same constant acceleration, g ≈ 9.8 m/s 2, (in a vacuum) independent of mass, size, shape, etc. Equations of Motion The acceleration due to gravity is almost constant and equal to 9.8 m/s 2 provided h << R E !

R. Field 1/22/2013 University of Florida PHY 2053Page 4 Example: Projectile Motion Near the Surface of the Earth In this case, a x = 0 and a y = -g, v x0 = v 0 cos , v y0 = v 0 sin , x 0 = 0, y 0 =h. (constant) Maximum Height H The time, t max, that the projective reaches its maximum height occurs when v y (t max ) = 0. Hence, For a fixed v 0 the largest H occurs when  = 90 o !

R. Field 1/22/2013 University of Florida PHY 2053Page 5 Exam 1 Spring 2011: Problem 4 A remote controlled toy car accelerates from rest at 2.0 m/s 2 under the power of its own wheels on a horizontal balcony until it shoots off the edge of the balcony 3.0 m from its starting point. The balcony is 10.0 m high. What is the horizontal distance from the point it left the balcony to where the car lands on level ground? Answer: 4.95 m % Right: 57% Let t h be the time the beanbag hits the ground. Let v d be the speed of the car when it shoots off the balcony. a = acceleration of car

R. Field 1/22/2013 University of Florida PHY 2053Page 6 Reference Frames Consider two frames of reference the O-frame (label events according to t,x,y,z) and the O'-frame (label events according to t',x',y',z') moving at a constant velocity V, with respect to each other at let the origins coincide at t= t' = 0. In the Galilean transformations the O and O' frames are related as follows: Galilean Velocity Transformation: Time is absolute! Classical velocity addition formula!

R. Field 1/22/2013 University of Florida PHY 2053Page 7 Postulates of Classical Physics First Postulate of Classical Physics (“Relativity Principle”): Second Postulate of Classical Physics (Galilean Transformation): The basic laws of physics are identical in all systems of reference (frames) which move with uniform (unaccelerated) velocity with respect to one another. The laws of physics are invariant under a change of inertial frame. The laws of physics have the same form in all inertial frames. It is impossible to detect uniform motion. The O and O' frame are related by the Galilean Transformation. Classical velocity addition formula!

R. Field 1/22/2013 University of Florida PHY 2053Page 8 Velocity Addition Theorem Velocity Addition Theorem (vector form): Example Problem: Jack wants to row directly across a river from the east shore to a point on the west shore. The width of the river is 250 m and the current flows from north to south at 0.61 m/s. The trip takes Jack 4.2 min. In what direction did he head his rowboat to follow a course due west across the river? At what speed with respect to the still water is Jack able to row? A relative to B B relative to C A relative to C North of West