Chapter 2 Kinematics in 1D Mechanics the study of the motion of objects atoms blood flow ice skaters cars planes galaxies Kinematics describes the motion of an object without reference to the cause of the motion Dynamics describes the effects that forces have on the motion of objects Chapter 5 Statics describes the effects that forces have on an object which is at rest bridge building Kinematics provides answers to the questions 1 Where is an object 2 What is its velocity 3 What is its acceleration y y A x x Displacement x xf xi displacement L 15 m 5 m 10 m in positive x direction If xf 5 m then x 5 m 5 m 10 m in positive x direction or 10 m in the negative x direction C 44 6 m 38 4 Speed and Velocity Average speed distance elapsed time D t While average velocity is displacement elapsed time v x avg x f x i x t f t i t L t time sec T time is a scalar in 1 dimension Or in vector notation A vector with dimensions of x f x i x L T in S I units are m v avg s t f ti t Simple Example A traveler arrives late at the airport at 1 08pm Her plane is scheduled to depart at 1 22pm and the gate is 2 1 km away What must be her minimum average running speed in m s to make the flight Solution Given ti 1 08 pm tf 1 22 pm D 2 1 km What is average speed vav t tf ti 1 22 1 08 14 mins vav D t 2 1 km 14 mins 0 15 km min 0 15 km min 1000 m 1 km 1 min 60 s vav 2 5 m s Instantaneous Speed and Velocity Instantaneous velocity is the velocity at some instant in time as t goes to zero Position X t x v x lim t 0 t Instantaneous speed is the magnitude of the instantaneous velocity Instantaneous velocity in vector notation ti Time x v lim t 0 t tf Acceleration The change in instantaneous velocity of an object gives the average acceleration ax avg v xf v xi t f ti Instantaneous acceleration v x ax lim t 0 t L T2 In S I units are m s2 We will mostly consider constant accelerations Equations of Kinematics Starting with the definitions of displacement velocity and acceleration we can derive equations that allow us to predict the motion of an object We will consider only constant acceleration Acceleration ax avg v xf v xi t f t i ax Solve for v ax t f t i v xf v xi v xf v xi ax t f t i Equation 2 7 q Velocity v x avg x f x i t f t i Solve for final position v x avg t f t i x f x i x f x i v x avg t f t i Vxf Vx vxi velocity time ti tf What is average velocity If acceleration is constant the average velocity the mean of the initial and final C 44 6 m is38 4 velocity 1 v x avg v xi v xf 2 1 x f xi v xi v xf t f ti 2 Equation 2 10 Also can substitute in the velocity v to give 1 x f xi v xi v xi a x t f ti t f ti 2 Or 1 2 x f xi v xi t f ti a x t f ti 2 Equation 2 11 What if we have no information about time It can be removed from the equations From acceleration equation t f ti v xf v xi ax Substitute into x equation v xf v xi 1 x f xi v xi v xf 2 ax 2 2 v xf v xi x f xi 2a x v 2 5 m s Since 2 xf 2 xi v xi v xf v xf v xi v v Then solving for vxf gives 2 xf 2 xi v v 2a x x f xi Equation 2 12 Summary Equations of Kinematics 1 x f xi v xi v xf t f ti 2 1 2 x f xi v xi t f ti a x t f ti 2 v xf v xi a x t f ti 2 xf 2 xi v v 2a x x f xi Note that the book used 0 instead of i and drops the f Example Problem A car is traveling on a dry road with a velocity of 32 0 m s The driver slams on the brakes and skids to a halt with an acceleration of 8 00 m s2 On an icy road the car would have skidded to a halt with an acceleration of 3 00 m s2 How much further would the car have skidded on the icy road compared to the dry road Solution Given vi 32 0 m s in positive x direction adry 8 00 m s2 in positive x direction aicy 3 00 m s2 in positive x direction Also vf 0 assume ti 0 xi 0 Find xdry and xicy or xicy xdry Example Problem A Boeing 747 Jumbo Jet has a length of 59 7 m The runway on which the plane lands intersects another runway The width of the intersection is 25 0 m The plane accelerates through the intersection at a rate of 5 70 m s2 and clears it with a final speed of 45 0 m s How much time is needed for the plane to clear the intersection Solution Given a 5 70 m s2 in x direction vf 45 0 m s in x direction Lplane 59 7 m Lintersection 25 0 m Assume ti 0 when nose of Jet enters intersection Find tf when tail of Jet clears intersection Example Problem you do An electron with an initial speed of 1 0x104 m s enters the acceleration grid of an old picture tube with a width of 1 0 cm It exits the grid with a speed of 4 0x106 m s What is the acceleration of the electron while in the grid and how long does it take for the electron to cross the grid Solution Given vi 1 0x104 m s in x direction vf 4 0x106 m s in x direction x 1 0 cm Find a 8 0x1014 m s2 and tf 5 0 ns Motion in Free fall q Consider 1D vertical motion on the surface of a very massive object Earth other planets the sun even large asteroids q Replace x with y in 1D kinematic equations q Acceleration is always non zero but constant q Acceleration of an object is due to gravity we will study gravitational forces later q All objects near the surface of the Earth experience the same constant downward acceleration q The acceleration due to gravity does not depend on the mass size shape density or any intrinsic property of …
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