l The study of HOW objects move: è Graphs è Equations è Motion maps è Verbal descriptions Kinematics-1

Overview l Kinematics: Description of Motion è Position and displacement è velocity »average »instantaneous è acceleration »average »instantaneous 10

Position vs Time Plots l Gives location at any time. l Displacement is change in position. l Slope gives velocity. 12 x (m) t Write a description of this motion. The object accelerates for the first 1 second, moving 3 m in that time. Then it slows down until its velocity is 0 at the 4 m mark. At the 4 m mark, it begins to speed up in the negative direction until it is at the 2 m mark at 2 s.

Position vs Time Plots l Gives location at any time. l Displacement is change in position. l Slope gives velocity. 12 x (m) t Position at t=3, x(3) = Displacement between t=1 and t=3.  x = Average velocity between t=1 and t=3 v = m m = -1.0 m -1 m / 2 s = -0.5 m/s  always means final - initial

Instantaneous and Average l Instantaneous Speed - speed at any given instant in time; speedometer reading, for example l Average Speed – total distance divided by total time of travel; rate you would have to travel constantly to cover the same distance in the same time

1.5 6 Velocity vs Time Plots l Gives velocity at any time. l Area gives displacement l Slope gives acceleration. 18 v (m/s) t Velocity at t=2, v(2) = Change in v between t=3 and t=5.  v = Average acceleration between t=5 and t=3: a = Displacement between t=0 and t=3: Average velocity between t=0 and t=3? 3 m/s 7.5 m t=0 to t=1: ½ (3m/s) (1 s) = 1.5 m 7.5 m / 3s = 2.5 m/s -2 m/s – 3 m/s = -5 m/s -5 m/s / (2 s) = -2.5 m/s 2 t=1 to t=3: (3m/s) (2 s) = 6 m

1.5 6 Velocity vs Time Plots l Gives velocity at any time. l Area gives displacement l Slope gives acceleration. 18 v (m/s) t Write a description of this motion. The object accelerates from 0 m/s to 3 m/s during the first second. Then for 2 seconds, it travels at a constant velocity of 3 m/s. From 3 seconds to 5 seconds, it accelerates in the negative direction from 3 m/s to –2 m/s.

2 6 Acceleration vs Time Plots l Gives acceleration at any time. l Area gives change in velocity 21 a (m/s 2 ) t Acceleration at t=4, a(4) = Change in v between t=4 and t=1.  v = -2 m/s 2 t=1-3:  v = (3m/s 2 )(2s) = 6 m/s t=3-4:  v = (-2m/s 2 )(1s) = -2 m/s +4 m/s

Where is velocity zero? Where is velocity positive? Where is velocity negative? Where is speed largest ? Where is acceleration zero? Where is acceleration positive ? position vs. time velocity vs. time Example 48

Summary of Concepts l kinematics: A description of motion l position: your coordinates l displacement:  x = change of position l velocity: rate of change of position è average :  x/  t è instantaneous: slope of x vs. t l acceleration: rate of change of velocity è average:  v/  t è instantaneous: slope of v vs. t 50

Velocity l Slope of x vs t gives v vs t Area under v vs t gives  x! l Which plot best represents v(t) x(t) tt xx v(t) t t tt 5s 4s

Acceleration (m/s 2 ) l The average acceleration is the change in velocity divided by the change in time. Instantaneous acceleration is limit of average velocity as  t gets small. It is the slope of the v(t) plot t v(t) tt vv t 31

Speeding Up & Slowing Down Negative acceleration can mean speeding up or slowing down. The same is true with positive acceleration.

Instantaneous Velocity l Instantaneous velocity is the limit of the average velocity as the time interval approaches 0. l It is the instantaneous rate of change of position with time. l Instantaneous rate of change is the derivative – which means slope of a graph at a pointslope of a graph at a point

Acceleration l Acceleration is a vector quantity defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.vector quantityvelocity

Acceleration Slope of V vs T – 1 Time (s)Velocity(m/s)

Acceleration Slope of V vs T -- 2 Time (s)Velocity(m/s)

Constant Positive Velocity

Constant Negative Velocity

Position – Time Constant Acceleration

Positive Velocity Positive Acceleration

Positive Velocity Negative Acceleration

Negative Velocity Negative Acceleration

Negative Velocity Positive Acceleration

l When does the red car pass the blue car? l Where does the red car pass the blue car? l How fast is each traveling at that time? Passing Lane

Passing Lane Velocity-Time

Passing Lane – Position-Time