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2-d Motion: Constant Acceleration

Learn about the kinematic equations of motion for 2-dimensional constant acceleration, as well as the effects of gravity on objects near the Earth's surface. Includes examples and reference frames.

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2-d Motion: Constant Acceleration

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  1. 2-d Motion: Constant Acceleration • Kinematic Equations of Motion (Vector Form) Acceleration Vector (constant) Warning! These equations are only valid if the acceleration is constant. Velocity Vector (function of t) Position Vector (function of t) The velocity vector and position vector are a function of the time t. Velocity Vector at time t = 0. Position Vector at time t = 0. The components of the acceleration vector, ax and ay, are constants. The components of the velocity vector at t = 0, vx0 and vy0, are constants. The components of the position vector at t = 0, x0 and y0, are constants. PHY 2053

  2. 2-d Motion: Constant Acceleration • Kinematic Equations of Motion (Component Form) Warning! These equations are only valid if the acceleration is constant. constant constant The components of the acceleration vector, ax and ay, are constants. The components of the velocity vector at t = 0, vx0 and vy0, are constants. The components of the position vector at t = 0, x0 and y0, are constants. • Ancillary Equations Valid at any time t PHY 2053

  3. Acceleration Due to Gravity y-axis h x-axis RE Earth • Experimental Result Near the surface of the Earth all objects fall toward the center of the Earth with the same constant acceleration, g ≈ 9.8 m/s2, (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/s2 provided h << RE! PHY 2053

  4. Example: Projectile Motion • Near the Surface of the Earth In this case, ax= 0 and ay= -g, vx0 = v0cosq, vy0 = v0sinq, x0 = 0, y0 =h. (constant) • Maximum Height H The time, tmax, that the projective reaches its maximum height occurs when vy(tmax) = 0. Hence, For a fixed v0 the largest H occurs when q = 90o! PHY 2053

  5. Exam 1 Spring 2011: Problem 4 • A remote controlled toy car accelerates from rest at 2.0 m/s2 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? a = acceleration of car Let vd be the speed of the car when it shoots off the balcony. Answer: 4.95 m % Right: 57% Let th be the time the beanbag hits the ground. PHY 2053

  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: Time is absolute! Classical velocity addition formula! • Galilean Velocity Transformation: PHY 2053

  7. Postulates of Classical Physics • First Postulate of Classical Physics (“Relativity Principle”): 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. • Second Postulate of Classical Physics (Galilean Transformation): The O and O' frame are related by the Galilean Transformation. Classical velocity addition formula! PHY 2053

  8. Velocity Addition Theorem • Velocity Addition Theorem (vector form): A relative to C B relative to C A relative to B • 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? North of West PHY 2053

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