Chapter 2 One Dimensional Kinematics Learning Objectives Position, Distance, and Displacement Average Speed and Velocity Instantaneous Velocity Acceleration Motion with Constant Acceleration Applications of the Equations of Motion Freely Falling Objects

Position, Distance, and Displacement Coordinate system  defines position Distance  total length of travel (SI unit = meter, m) Scalar quantity Displacement  change in position Change in position = final pos. – initial pos.  x = xf – xi Vector quantity

Position, Distance, and Displacement Before describing motion, you must set up a coordinate system – define an origin and a positive direction. The distance is the total length of travel; if you drive from your house to the grocery store and back, what is the total distance you traveled? Displacement is the change in position. If you drive from your house to the grocery store and then to your friend’s house, what is your total distance? What is your displacement?

Average speed and velocity Average speed  distance traveled divided by the total elapsed time SI units, meters/second (m/s) Scalar quantity Always positive

Average speed and velocity Average velocity  displacement divided by the total elapsed time SI units of m/s Vector quantity Can be positive or negative

Graphical Interpretation of average velocity What’s the average velocity between the intervals t = 0 s  t = 3 s? Is the particle moving to the left or right? What’s the average velocity between the intervals t = 2 s  t = 3 s? Is the particle moving to the left or right?

Instantaneous Velocity This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous velocity. Magnitude of the instantaneous velocity is known as the instantaneous speed

Instantaneous velocity If we have a more complex motion This plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity is tangent to the curve.

Acceleration Acceleration – how fast you speed up, slow down, or change direction; it’s the rate at which velocity changes. Two examples: t (s) v (mph) 55 1 57 2 59 3 61 t (s) v (m/s) 34 1 31 2 28 3 25 a = -3 m / s s = -3 m / s 2 a = +2 mph / s

Acceleration Average acceleration  the change in velocity divided by the time it took to change the velocity SI units meters/(second · second), m/s2 Vector quantity Can be positive or negative Accelerations give rise to force

Acceleration Instantaneous acceleration - This means that we evaluate the average acceleration over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous acceleration. When acceleration is constant, the instantaneous and average accelerations are equal

Acceleration due to Gravity Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance). This acceleration vector is the same on the way up, at the top, and on the way down! a = -g = -9.81 m/s2 9.81 m/s2 Interpretation: Velocity decreases by 9.81 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.

Graphical interpretation of Acceleration Velocity vs time (v-t) graphs give us information about: average acceleration, instantaneous acceleration average velocity  slope of a line on a x-t plot is equal to the average velocity over that interval the “+” 0.25 m/s2 means the particle’s speed is increasing by 0.25 m/s every second What does the “-” 0.5 m/s2 mean?

Motion with constant acceleration If the acceleration is constant, the velocity changes linearly: Average velocity: v0 = initial velocity a = acceleration t = time Can you show that this equation is dimen- sionally correct? v  t

Velocity & Acceleration Sign Chart + - Moving forward; Speeding up Moving backward; Slowing down Moving forward; Slowing down Moving backward; Speeding up

Acceleration Under which scenarios does the car’s speed increase? Decrease?

Kinematics Formula Summary For 1-D motion with constant acceleration: vf = v0 + a t vavg = (v0 + vf ) / 2 x = v0 t + ½ a t 2 vf2 – v02 = 2 a x (derivations to follow)

Kinematics Derivations a = v / t (by definition) a = (vf – v0) / t  vf = v0 + a t vavg = (v0 + vf ) / 2 will be proven when we do graphing. x = v t = ½ (v0 + vf) t = ½ (v0 + v0 + a t) t  x = v0 t + a t 2 (cont.)

Kinematics Derivations (cont.) vf = v0 + a t  t = (vf – v0) / a x = v0 t + a t 2  x = v0 [(vf – v0) / a] + a [(vf – v0) / a] 2  vf2 – v02 = 2 a x Note that the top equation is solved for t and that expression for t is substituted twice (in red) into the x equation. You should work out the algebra to prove the final result on the last line.

Sample Problems You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration? -15.6 m/s2