How things Move Ancient Greek philosopher and scientist Aristotle developed the earliest theory of how things move. natural motion – motion that could maintain itself without the aid of an outside agent. (Pushing a rock off the ledge, falls to the ground) liquids falling or running downhill, air rising, flames leaping upward Aristotle believed everything was made of four elements Fire Air Water Earth Aristotle's Periodic Table “Natural Motion” (vertical) “Violent Motion” (horizontal)

How things Move Fire Air Water Earth Aristotle's Periodic Table “Natural Motion” (vertical) “Violent Motion” (horizontal) Each element has its own natural motion, and its own place that it strives to be Aristotle believed an objects natural motion was determined by how much of each element the object contained (rock sink in water because it contained mostly earth, wood floated because it contained mostly air) earth moves downward because Earth’s center is it’s natural resting place water’s natural resting place is on top of earth Violent Motion – motion that forced objects to behave contrary to an objects natural motion, meaning an external push or pull was needed

How things Move Aristotle believed that all motion on Earth was either “natural” or “violent” Motion not on earth followed a different set of rules 5 th element – ether (from the Greek word for to kindle or blaze) – had no weight and was unchangeable, and perfect in every way moon, sun, planets and stars were made of ether celestial motion “perfect circles” ether’s natural place was in the “heavens” and it moved in perfect circles object’s on earth could not move the way the star’s did because they did not contain ether Aristotle's physics governed science until about the mid 16 th century Popular because it reinforced religious beliefs “………fuse another five elements…” – Wu-Tang Clan

In the seventeenth century Newton developed Calculus which changed the way we think about motion. Describing MotionDisplacement Instantaneous Speed Average Speed, velocity Velocity Acceleration Quantities which characterize motion Displacement – the change from one position, x 1 to another position x 2 Greek letter, “delta”, mathematically means, “the change in”. Displacement is a vector quantity – a vector has both size (aka magnitude) and direction If I start at a position of –2 m, and end at a position of 3 m, what is my displacement?

Average velocity the ratio of the displacement,  x, that occurs during a particular time interval,  t. Examples of speed: 55 mi/hr, 20 m/s, 300 km/hr speed – 1 piece of info Examples of velocity: 55 mi/hr, due West 20 m/s, straight up 300 km/hr, 37 degrees East of North North South East West velocity – 2 pieces of info Speed – answer the question: “How fast?” Velocity – answers the questions: “How fast and in what direction am I traveling?” Quantities which only have magnitude or size are known as scalars Quantities that have both magnitude and direction are known as vectors NOTE: We usually call t i the starting time and set it equal to 0, t i =0

Average Velocity: This is a position versus time plot. From here what is the average velocity from t = 0 to t = 3s? The slope of this line is V avg !

The slope of the line gives us information on the direction of the velocity? + slope = + displacement - slope = - displacement

Average speed, s AVG, is a scalar quantity GO TO HITT QUESTION Instantaneous Velocity and Speed If we want to know the velocity of a particle at an instant we simply obtain the average velocity by shrinking the time interval  t closer and closer to 0.

start finish d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 ad infinitum the trip is built out of an infinitely large number of points just like this one instantaneous speed at this location zoom-in

Example: If a particle’s position is given by x = 4-12t+3t 2, what is its velocity at t = 1s? Evaluate this as t = 1 s gives us: What does the “-”, negative sign mean? This tells us the direction of the particle at time t = 1s.

Acceleration: When a particle experiences a change in velocity is undergoes an acceleration. The average acceleration over a time interval is defined as: The instantaneous acceleration is the derivative of velocity with respect to time: The acceleration of a particle at any time is the second derivative of it’s position with respect to time. NOTE: acceleration is a vector quantity

Typical accelerations: Ultracentrifuge3 x 10 6 m/s 2 Batted baseball3 x 10 4 m/s 2 Bungee Jump30 m/s 2 Acceleration of gravity on Earth9.81 m/s 2 Emergency stop in a car8 m/s 2 Acceleration of gravity on Moon1.62 m/s 2 Note:Acceleration of gravity on Moon Acceleration of gravity on Earth = =  16.5 %

slope of v(t) velocity at any time can be found from the slope of the x(t) graph acceleration at any time can be found from the slope of the v(t) graph

in most cases the acceleration is constant: Example: Car skidding, free falling objects, etc. When the acceleration is constant, the average acceleration and instantaneous acceleration are equal so we have: Multiplying both sides by t: NOTE: Check if this is correct, what is the final velocity equal to at t = 0. V i !! YEAH!

Check this out: Remember that: If we set t i = 0 and t f = t we can rewrite this to be: multiplying each side by t and rearrange to x f : it turns out that for a particle experiencing constant acceleration: now plug this into this equation now from before: plug this into here

we just call this the distance, d, traveled next we can rearrange this equation: time, t: to solve for, now substitute this equation into here

Equation: Do this for homework! Equation summary: EquationMissing Quantity v f t a v i

Example: On a dry road, a car with good tires may be able to brake with a constant deceleration of 4.9s m/s 2. How long does it take the car, initially traveling at 24.6 m/s, take to stop? Given:a = -4.9 m/s 2 v i = 24.6 m/s v f = 0 m/s Find:t = ? Objects that undergo free fall are just a case of a particle under constant acceleration.

Aristotelian physics had a short coming what is I had a rock with some weight. And I had a container of water, same size, shape and weight? In fact all falling objects fall at the same rate, called the acceleration of gravity (neglecting air resistance) Drop different objects their speed will increase at the same rate! Their speed will increase by ~ 10 m/s (32 ft/s) every second Free - Fallin

Free Fall Measurement –important ratios Time DistanceSpeedTotal Distance 1 5 m10 m/s5 m Let 1 unit of distance = the distance the object falls during the first second. This turns out to be 4.9 m ~ 5 m The acceleration is uniform, g = 9.8 m/s/s ~ 10 m/s/s 1

Free Fall Measurement Time DistanceSpeedTotal Distance 1 5 m10 m/s5 m 2 15 m20 m/s20 m 1 3

Free Fall Measurement Time DistanceSpeedTotal Distance 1 5 m10 m/s5 m 2 15 m20 m/s20 m 3 25 m30 m/s45 m 1 3 5

Free Fall Measurement Time DistanceSpeedTotal Distance 1 5 m10 m/s5 m 2 15 m20 m/s20 m 3 25 m30 m/s45 m 4 35 m40 m/s80 m

Free Fall Measurement Time DistanceSpeedTotal Distance 1 5 m10 m/s5 m 2 15m20 m/s20 m 3 25m30 m/s45 m 4 35m40 m/s80 m 5 45m50 m/s125 m

Free Fall Measurement Time DistanceSpeedTotal Distance 1 5 m10 m/s5 m 2 15 m20 m/s20 m 3 25 m30 m/s45 m 4 35 m40 m/s80 m 5 45 m50 m/s125 m … t t speed of descent ~ time of fall distance of fall ~ (time of fall) 2

time (s)speed (m/s) holy smokes!! that’s g speed of descent ~ time of fall

time (s)distance (m) distance of fall ~ (time of fall) 2