Kinematics II January 14 2019 Remember this graph A 60m 30m 30m v 3 75m s 8 sec 8s D 0m 45m v 4 5m s 10 s WHAT ABOUT B Conclusion Velocity is not the same at all times It is changing A changing velocity is called an acceleration aaverage v final vinitial t dv a dt v t Acceleration in Pictures Constant Velocity Constant Acceleration Average Acceleration Acceleration is the rate of change of the velocity Dimensions are L T2 SI units are m s Example Constant acceleration a 1 m s2 Time s Velocity m s 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 v a t v at v f vinitial v f v0 at Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as t approaches 0 2 Dv x dv x d x a x lim 2 Dt 0 Dt dt dt Calculus dx v dt 2 dv d x a 2 dt dt Instantaneous Acceleration graph The slope of the velocity vs time graph is the acceleration The green line represents the instantaneous acceleration The blue line is the average acceleration Acceleration and Velocity 1 When an object s velocity and acceleration are in the same direction the object is speeding up When an object s velocity and acceleration are in the opposite direction the object is slowing down Example Acceleration and velocity are in opposite directions Acceleration is uniform blue arrows maintain the same length Velocity is decreasing red arrows are getting shorter Positive velocity and negative acceleration Some math Assume constant acceleration dv dt constant a dv adt vf dv v v0 t f t v0 adt a dt a t 0 at 0 0 v f v0 at v v0 at v v0 at More dx v0 at dt dx v0 dt atdt x t t dx v dt a tdt 0 x0 0 0 1 2 x x0 v0t at 2 1 2 x x0 v0t at 2 One more calculus trick the dv chain rule dv dv dx a v dt dx dt adx vdv dx vf x a dx vdv x0 v0 1 2 2 a x x0 v f v0 2 for x 0 0 2 f 2 0 v v 2ax Kinematic Equations summary from the book Kinematic Equations The kinematic equations may be used to solve any problem involving one dimensional motion with a constant acceleration You may need to use two of the equations to solve one problem Many times there is more than one way to solve a problem Kinematic Equations specific For constant a Can determine an object s velocity at any time t when we know its initial velocity and its acceleration Does not give any information about displacement Kinematic Equations specific For constant acceleration v xi v xf vx 2 The average velocity can be expressed as the arithmetic mean of the initial and final velocities Kinematic Equations specific For constant acceleration 1 2 x f xi v xi t a x t 2 Gives final position in terms of velocity and acceleration Doesn t tell you about final velocity Graphical Look at Motion displacement time curve The slope of the curve is the velocity The curved line indicates the velocity is changing Therefore there is an acceleration Graphical Look at Motion velocity time curve The slope gives the acceleration The straight line indicates a constant acceleration Graphical Look at Motion acceleration time curve The zero slope indicates a constant acceleration Let s get real and throw something out of a building and watch it fall This is called free fall Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity alone It does not depend upon the initial motion of the object Dropped released from rest Thrown downward Thrown upward Acceleration of Freely Falling Object The acceleration of an object in free fall is directed downward regardless of the initial motion The magnitude of free fall acceleration is g 9 80 m s2 g decreases with increasing altitude g varies with latitude 9 80 m s2 is the average at the Earth s surface In the English American system g 32 ft sec2 Acceleration of Free Fall Rise cont We will neglect air resistance Free fall motion is constantly accelerated motion in one dimension Use the kinematic equations to solve problems Free Fall Example Let s drop an object from the top of this building by simply releasing it How long will it take to get to the ground How fast will it be going when it gets to the ground What color is the object Issues Where is the origin IS y the same thing as x How can that be Use Transparencies Motion Equations from Calculus Displacement equals the area under the velocity time curve lim Dtn 0 v n tf xn Dtn v x t dt ti The limit of the sum is a definite integral
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