Math Smack

Kinematics: Velocity, Speed, and Rates of Change Greg Kelly, Hanford High School, Richland, Washington

Rectilinear Motion Motion on a line Suppose a dog moves along a straight line so that its position s from an origin O is given as some function of time t. We write s = s(t) where t > 0. Moving in a positive direction from the origin Moving in a negative direction from the origin

Position Function Horizontal axis: – time Vertical Axis: – position on a line Moving in a positive direction from the origin time position Moving in a negative direction from the origin Position function: s(t) s = position (position) t = time s(t)= position changes as time changes

Example Use the position and time graph to describe how the puppy was moving time position

Distance =Displacement = Change in position is measured in both displacementdistanceand If you were to drive 10 miles eastand then 4 miles west 10 miles east 4 miles west The difference between the starting and ending positions of an object The total amount of “ground covered” by an object or the total length of its path 14 miles6 miles

position time Distance =Displacement = And the position graph would look like this: 10 miles east4 miles west 14 miles6 miles

Velocity Rate – position change vs time change – Velocity can be positive or negative positive: going in a positive direction negative: going in a negative direction Velocity Position

Velocity function Velocity is the slope of the position function (change in position /change in time) velocity = – Technically this is instantaneous velocity PositionVelocityMeaning Positive SlopePositive y’smoving in a positive direction Negative slopeNegative y’sMoving in a negative direction

Velocity Rate at which a coordinate of a particle changes with time – Insanities velocity s(t) = position with respect to time Instantaneous velocity at time t is: – time position v(t) = positive – increasing slope – moving in a positive direction v(t) = negative – decreasing slope – moving in a negative direction

Practice Let s(t)= t 3 -6t 2 be the position function of a particle moving along an s-axis were s is in meters and t is in seconds. – Graph the position function – On a number line, trace the path that the particle took. – Where will the velocity be positive? Negative? – Graph the instantaneous velocity. – Identify on the velocity function when the particle was heading in a positive direction and when it was heading in a negative direction.

Practice s(t)= t 3 -6t 2 The velocity and speed functions are given by: Speed is the absolute value of velocity

Velocity or Speed Speed is the change in position with respect to time in any direction Velocity is the change in position with respect to time in a particular direction – Thus – Speed cannot be negative – because going backwards or forwards is just a distance – Thus – Velocity can be negative – because we care if we go backwards

Speed Absolute Value of Velocity – example: if two particles are moving on the same coordinate line with velocity of v = 5 m/s and v= -5 m/s, then they are going in opposite directions but both have a speed of |v| = 5 m/s

Where will the velocity be positive? Negative? s(t)= t 3 - 6t 2

Where will the velocity be positive? Negative? s(t)= 3t t

This position graph tells us that the particle is on the negative side of the origin form 0 6 and is at the origin at times t = 0 and t = 6 s(t)= t 3 - 6t 2

s’(t)= 3t t The velocity graph tells us that the particle is moving in the negative direction if 0 4, and is momentarily stopped at t = 0 and t = 4 (the velocity is zero at those times)

The speed graph tells us that the speed of the particle is increasing for 0 4.

Example - s(t)= t 3 - 6t 2 position time velocity time speed

Velocity is the first derivative of position. Imagine Wentz falls from a 196 foot platform can be expressed with the equation: …where t is in seconds and s is measured in feet. Given the above statement, the equation for his velocity is: …which is expressed in what units? feet/second

Find Which is easy enough except…  32 what? t (seconds) v(t) (feet/second) First, let’s look at the graph of velocity. Note that like the position graph, s ( t ), the y-axis represents velocity while the x axis represents time

Acceleration feet/second 2 Since the slope of a line is based on: …or in this case: …we are now talking about: Is an expression of… which is a rate of change of velocity. Acceleration

Acceleration is the derivative of velocity. feet/second 2 Remember: If you ever get lost, what can always save you?UNITS!

Example: Free Fall Equation Speed is the absolute value of velocity. Gravitational Constants:

Wait! So what is the difference between speed and velocity? Speed is the absolute value of velocity. If the object is moving upward… …or to the right If the object is moving downward… …or to the left But speed does not indicate direction so speed will always be positive…

Wait! So what is the difference between speed and velocity? Speed is the absolute value of velocity. If an object is moving upward at a speed of 40 ft/sec… If an object is moving downward at a speed of 40 ft/sec But regardless of the direction, the speed will always be positive…

Interpret the signs

Analyzing Motion GraphicallyAlgebraicallyMeaning Position Velocity  Acceleration Positive “s” values Positive side of the number line Negative side of the number line Negative “s” values s  (t)=velocity. Look for Critical Pts Postive “v” values 0 “v” values (CP) Negative “v” values Moving in + direction Turning/stopped Moving in a – direction v  (t)= acceleration Look for Critical Pts + a, + v = speeding up - a, - v = speeding up + a, - v = slowing down - a, + v = slowing down

An often confused quantity, acceleration has a meaning much different than the meaning associated with it by sports announcers and other individuals. The definition of acceleration is: Acceleration is a vector quantity that is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.vector quantityvelocity

Sports announcers will say that a person is accelerating if he/she is moving fast. Yet acceleration has nothing to do with going fast. A person can be moving very fast and still not be accelerating. Acceleration has to do with changing how fast the velocity of an object is changing. If an object is not changing its velocity, then the object is not accelerating. Click HereHere

Positive velocity means moving forward that is change in position increases /per time. Negative velocity means moving backwards, that is, change in position decreases per time

Positive acceleration means velocity (remember that velocity = |speed|) is increasing; that is, you are going faster as time moves forward Negative acceleration means you are slowing down, that is, you are going slower as time goes by--but this is tricky with negative velocities...

If a car is moving backwards at a certain rate of speed, and moves backwards faster, that is, it goes from -50 mph to -70 mph, then it is negative acceleration. The rate of speed is increasing, because it is a backwards direction, it is really decreasing. You have to remember that velocity deals with both "amount" of speed and direction. I find it very helpful to draw a graph and estimate the slope. Remember that the slope of the graph of velocity vs. time will be acceleration.

Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t t t +3 Analyze the motion of the particle for t > 0 GraphicallyAlgebraicallyMeaning  Position Velocity Acceleration Never 0 (t > 0), always positive Always on postive side of number line < t < 2 going pos dir. t = 2 turning 2 < t < 5 going neg. dir t=5 turning t > 5 going pos. direc t=0 t=2 t= v a <t<2 slowing down 2 < t < 3.5 speeding up 3.5 < t < 5 slowing down 5 < t speeding up

Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t t t + 3 Analyze the motion of the particle for t > 0 position velocity Acceleration

position velocity Acceleration Position t Direction of motion stop positive Dir. negative Dir positive Dir v(t) /2 a(t) slowing down speeding up slowing down speeding up

Homework Chapter 17 A.1 (All) Chapter 17 A.2 (#1 – #3) Random students will be graded on this. Be ready. No late or revisions will be accepted.

Chapter 17 A.2 # 4

Chapter 17 A.2 #

t=0

Chapter 17 A.2 # VelocityAcceleration ++

If a car is moving backwards at a certain rate of speed, and moves backwards faster, that is, it goes from -50 mph to -70 mph, then it is negative acceleration. The rate of speed is increasing, because it is a backwards direction, it is really decreasing. You have to remember that velocity deals with both "amount" of speed and direction. I find it very helpful to draw a graph and estimate the slope. Remember that the slope of the graph of velocity vs. time will be acceleration.

Steepness of slope on Position-Time graph Slope is related to velocity Steep slope = higher velocity Shallow slope = less velocity

Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t t t +3 Analyze the motion of the particle for t > 0 GraphicallyAlgebraicallyMeaning  Position Velocity Acceleration Never 0 (t > 0), always positive Always on postive side of number line < t < 2 going pos dir. t = 2 turning 2 < t < 5 going neg. dir t=5 turning t > 5 going pos. direc t=0 t=2 t= v a <t<2 slowing down 2 < t < 3.5 speeding up 3.5 < t < 5 slowing down 5 < t speeding up

Different Position. Vs. Time graphs Constant positive velocity (zero acceleration) Constant negative velocity (zero acceleration) Increasing positive velocity (positive acceleration) Decreasing negative velocity (positive acceleration) Uniform Motion Accelerated Motion

Different Position. Vs. Time Changing slope means changing velocity!!!!!! Decreasing negative slope = ?? Increasing negative slope = ??

X t A B C A … Starts at home (origin) and goes forward slowly B … Not moving (position remains constant as time progresses) C … Turns around and goes in the other direction quickly, passing up home

Graphing w/ Acceleration x A … Start from rest south of home; increase speed gradually B … Pass home; gradually slow to a stop (still moving north) C … Turn around; gradually speed back up again heading south D … Continue heading south; gradually slow to a stop near the starting point t A B C D

Tangent Lines t SLOPEVELOCITY Positive Negative Zero SLOPESPEED SteepFast GentleSlow FlatZero x On a position vs. time graph:

Increasing & Decreasing t x Increasing Decreasing On a position vs. time graph: Increasing means moving forward (positive direction). Decreasing means moving backwards (negative direction).

Concavity t x On a position vs. time graph: Concave up means positive acceleration. Concave down means negative acceleration.

Special Points t x P Q R Inflection Pt. P, R Change of concavity, change of acceleration Peak or ValleyQ Turning point, change of positive velocity to negative Time Axis Intercept P, S Times when you are at “home”, or at origin S

Acceleration vs. Time Time is on the x-axis Acceleration is on the y-axis Shows how acceleration changes over a period of time. Often a horizontal line.

All 3 Graphs t Position Velocity t acceleration t

Real life a t P t Note how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the orange segments would be connected. In our class, however, we’ll only deal with constant acceleration.

Constant Rightward Velocity

Constant Leftward Velocity

Constant Rightward Acceleration

Constant Leftward Acceleration

Leftward Velocity with Rightward Acceleration

Yo-Yo a Go-Go: Position, Velocity, and Acceleration None of your friends, if you have any, just kidding, will ever complain or even notice if you use the words “velocity” or “speed” interchangeably. But your friendly, neighborhood mathematician will With a yo-yo, upward motion is positive velocity. Downward motion is negative velocity. Speed is always positive (or zero).

I Like Yo-Yos Using these three functions and their graphs, I want to discuss several things about the yo-yo’s motion. H(t) = t 3 – 6t 2 + 5t What does mean? V(t) = H’(t) = 3t t + 5 What does mean? A(t) = V’(t) = H”(t) = 6t – 12 What does mean?

H(t) = t 3 – 6t 2 + 5t + 30 Endpoints: (0, 30) & (4, 18)

V(t) = H’(t) = 3t t + 5 Endpoints: (0, 6) & (4, 5)

A(t) = V’(t) = H”(t) = 6t – 12 Endpoints (0, -12) & (4, 12)

Maximum and Minimum Height Maximum and Minimum happen at local extrema. Look at the graphs and these are the highest and lowest points on the interval [0,4] To locate them, take the derivative of H(t). That’s V(t). Set it equal to zero and solve. V(t) = H’(t) = 3t t + 5 = 0 You may need the Quadratic Equation. I’ll wait so you can find an answer.

Maximum and Minimum Height

H(t) = t 3 – 6t 2 + 5t + 30

Maximum and Minimum Height The yo-yo gets as high as 31.1 inches above the ground at t ≈ 0.47 seconds. It gets as low as 16.9 inches above the ground at t ≈ 3.53 seconds.

Total Displacement Total displacement is FINAL POSITION minus INITIAL POSITION. A yo-yo starts at a height of 30 inches and ends at a height of 18 inches. So, the total displacement is: 18 – 30 = -12. Why does this have a negative answer? The net movement is DOWNWARD

Some Velocities…??? Average Velocity: Total Displacement ÷ elapsed time This will tell you the yo-yo is, on average, going down 3 inches per second. Maximum Velocity: Determine the yo-yo’s maximum velocity. Minimum Velocity: Determine the yo-yo’s minimum velocity.

Maximum and Minimum Velocity This is during the interval 0 to 4 seconds. Set the derivative of V(t), that’s A(t), equal to zero and solve. A(t) = V’(t) = 6t – 12 = 0. 6t – 12 = 0 6t = 12 t = 2.

V(t) = H’(t) = 3t t + 5 t = 2.

Look at the graph A(t). At t = 2, you get zero. V(t) = 3t t + 5 Check the endpoints and at t = 2 V(0) = 5  Maximum Velocity V(2) = -7  Minimum Velocity V(4) = 5  Maximum Velocity Maximum and Minimum Velocity