Chapter 2 Motion in One Dimension Motion an ingredient in everything in physics Depends on displacement velocity and acceleration 2 1 Displacement 2 1 Displacement Defined as the change in position x xf xi f stands for final and i stands for initial May be represented as y if vertical Units are meters m in SI centimeters cm in cgs or feet ft in US Customary 1 Displacement Isn t Distance Position of car at various times The displacement of an object is not the same as the distance it travels Example Throw a ball straight up and then catch it at the same point you released it The distance is twice the height The displacement is zero 2 Vector and Scalar Quantities Vector quantities need both magnitude size and direction to completely describe them number and sign Generally denoted by boldfaced type and an arrow over the letter or sign is sufficient for this chapter Moving truck A truck moves 70 m east then moves 120 m west and finally moves east again a distance of 90 m If east is chosen as the positive direction what is the truck s resultant displacement Remember Displacement isn t distance Scalar quantities are completely described by magnitude only only number 3 2 2 Velocity Speed and velocity The average speed of an object is defined as the total distance traveled divided by the total time elapsed Average speed v d t total distance total time Average speed totally ignores any variations in the object s actual motion during the trip The total distance and the total time are all that is important SI units are m s Speed is a scalar quantity 4 Velocity Velocity It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs vaverage x xf xi t tf ti generally use a time interval so let ti 0 Direction will be the same as the direction of the displacement time interval is always positive or is sufficient Units of velocity are m s SI cm s cgs or ft s US Cust Other units may be given in a problem but generally will need to be converted to these 5 Speed vs Velocity Graphical Interpretation of Velocity Velocity can be determined from a position time graph Average velocity equals the slope of the line joining the initial and final positions Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger An object moving with a constant velocity will have a graph that is a straight line 6 Constant Velocity The straight line indicates constant velocity The slope of the line is the value of the average velocity Non constant velocity The motion is nonconstant velocity The average velocity is the slope of the blue line joining two points 7 Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short or as the time interval approaches zero x v t lim0 t The instantaneous velocity indicates what is happening at every point of time your car s speedometer 2 3 Acceleration Changing velocity non uniform means an acceleration is present Acceleration is the rate of change of the velocity a v vf vi t tf ti Units are m s SI cm s cgs and ft s US Cust Vector quantity 8 Relationship Between Acceleration and Velocity Uniform velocity shown by red arrows maintaining the same size Acceleration equals zero Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform blue arrows maintain the same length Velocity is decreasing red arrows are getting shorter Velocity is positive and acceleration is negative 9 2 5 One dimensional motion with constant acceleration a constant Initial t 0 velocity v0 position x0 At time t velocity v position x a v v0 t Or v v0 v x x0 t Or at x x0 vt you can pick your coordinate system so that x0 0 Use Equations of Constant Acceleration v v0 a t x x0 v0 t 1 2 a t2 v2 v02 2 a x x0 In most cases x0 0 then there are 5 variables a acceleration t time v0 initial velocity v final velocity x displacement distance As long as you know any 3 of them You can find the rest by using the above three equations 10 y a 9 8 m s direction 2 6 Free Falling Bodies Free Fall an object dropped Acceleration is always g downwards Velocity may be positive zero or Position may be positive zero or g 9 8 m s a g vy vy0 gt Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is g 9 80 m s2 vo 0 a g y y0 vy0t 1 2 gt2 vy2 vy02 2g y y0 11 Free Fall an object thrown downward a g 9 80 m s2 Initial velocity 0 With upward being positive initial velocity will be negative Free Fall object thrown upward Initial velocity is upward so positive The instantaneous velocity at the maximum height is zero a g 9 80 m s2 everywhere in the motion v 0 12 Non symmetrical Free Fall Need to divide the motion into segments Possibilities include Upward and downward portions The symmetrical portion back to the release point and then the nonsymmetrical portion Free Fall Example Fred throws a ball 30 m s vertically upward How long does it take to hit the ground 2 meters below y y0 vy0t 1 2 gt2 where he released it y y0 vy0t g t2 y y0 vy0t g t2 0 2 30 t 9 8 t2 0 ax 2 bx c 0 b b 2 4ac x 2a vy vy0 gt vy2 vy02 2g y y0 2 30 302 4 9 8 2 x 9 8 t 6 19 s or 06 s 13