Lecture 2 (Walker: 2.1-2.3) Position, Displacement, Speed, and Velocity August 31, 2009

Some illustrations courtesy Prof. J.G. Cramer, U of Washington

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Physics Readiness Test • Results posted on course web page and on bulletin board across from Thornton 118 • “Pass” status -- Wait list students who passed will get an add permit • “Fail” status -- You will be dropped from Phys 111/112. • “ALEKS” status -- You will be dropped, but can get an add permit by achieving 80% or better proficiency in ALEKS (see course web page). Continue attending class and doing homework.

2 Scalar and Vector Quantities

• Scalar quantities are completely described by magnitude only (temperature, length,…) • Vector quantities need both magnitude (size) and direction to completely describe them (force, displacement, velocity,…) – Represented by an arrow; the length of the arrow is proportional to the magnitude of the vector – Head of the arrow shows the direction • Write vector as v r (or sometimes v)

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Temperature: Scalar Wind Velocity: Vector Quantity; specified by single Quantity; specified by its number giving its magnitude. magnitude & direction.

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Chapter 2 One-Dimensional (1-D) Kinematics One dimensional kinematics refers to motion along a straight line. Terms we will use: • Position, distance, displacement • Speed, velocity (average and instantaneous) • Acceleration (average and instantaneous)

6 Coordinate Systems A coordinate system is used to describe location, or position. A coordinate system consists of: – a fixed reference point called the origin (e.g., metal disk in street, center of table) – a set of axes and definition of “positive” directions (e.g., “x axis points East”) – the units for the axes (e.g., meters) The position of an object is its location in a coordinate system. Position is a vector quantity

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Cartesian coordinate system

• Also called rectangular coordinate system • x- and y- axes • position points are labeled (x,y)

The arrow on axis indicates the “positive” direction.

8 Plane polar coordinate system

– origin and reference line are noted – point is distance r from the origin in the direction of angle θ, ccw from reference line – position points are labeled (r,θ)

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SFSU: 37.72084N, -122.476619E

10 Position A • Position is defined in terms of a frame of reference (coordinate system)

• Frame A: xi >0 and xf >0

Frame B: x’i<0 but x’f >0 Note that we use subscripts to xi xf indicate different positions: B y’ xi initial position (or x0)

xf final position; x2 position #2 • Vector quantity; in 1-dim, usually use + or - to specify direction and write as just x (no arrow) x’ • SI Unit for position amount: x ’ O’ x ’ meter (m) i f 11

Displacement • Displacement measures the change in position ∆x – Represented as ∆x(if horizontal) or ∆y (if vertical)

∆x = x f − xi – Vector quantity; + or - generally sufficient to xi xf indicate direction for 1- dimensional motion SI Units: Meters (m) – Sometimes write as r r r ∆x = x f − xi

12 Distance Distance (scalar) is the total length of travel. SI unit: m If you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi.

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Displacement vs. Distance Displacement is the net change in position, and has a direction (maybe just + or - in 1-D). You drive from your house to the grocery store and then to your friend’s house, your net displacement is -2.1 mi:

∆x = x f − xi = 0 − 2.1mi = −2.1mi The distance you have traveled is 10.7 mi.

xf xi

14 Distance & Displacement?

• Distance may be, but is not necessarily, the magnitude of the displacement

Displacement Distance (orange line) (blue line)

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Position-time graphs

¾ Note: position-time graph is not necessarily a straight line, even though the motion is along x-direction

16 Average Speed The average speed (SI unit: m/s; scalar or vector?) is defined as the distance traveled divided by the time the trip took: Average speed = distance / elapsed time Is the average speed of the red car 40.0 mi/h, more than 40.0 mi/h, or less than 40.0 mi/h? Could average speed ever be negative?

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Average Velocity • Say takes time ∆t for an object to undergo a displacement ∆xr • The average velocity is rate at which the displacement occurs ∆xr xr − xr vr = = f i average ∆t ∆t • SI Unit: m/s • It is a vector; direction will be the same as the direction of the displacement (∆t is always positive) • + or - is sufficient direction description for 1-D motion; so ∆x x − x v = = f i average ∆t ∆t

18 Average Speed and Velocity Average velocity = displacement / elapsed time If you return to your starting point, your average velocity is zero. t=8s ∆x 50.0 m− 0 v == =6.25 m/s 8 s run ∆−t 8.0 s 0

∆x 0− 50.0 m v == =−1.25 m/s walk ∆−t 48.0 s 8.0 s

t=48s48 s ∆x 0 v == =0.0 m/s av ∆−t 48.0 s 0

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Average Speed and Velocity Avg. speed may be, but is not always, the magnitude of avg. velocity Graphical Interpretation of Average Velocity: The same motion, plotted one-dimensionally and as a two dimensional x-t graph:

Average speed (0-4s) = (7m)/(4s) = 1.75 m/s

Average velocity (0-4 s) = ?

20 Average Speed and Velocity Average speed may be, but is not necessarily, the magnitude of avg. velocity Graphical Interpretation of Average Velocity: The same motion, plotted one-dimensionally and as a two dimensional x-t graph:

Average speed (0-4s) = (7m)/(4s) = 1.75 m/s Average velocity (0-4 s) = (-2 m)/(4 s) = - 0.5 m/s Slope of line = average velocity

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Instantaneous Velocity ∆xr Definition: vr = lim ∆t→0 ∆t (2-4)

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. The instantaneous velocity gives the speed and direction of motion at each instant. What about instantaneous speed? Same as instantaneous velocity? What is the “speedometer” in a car measuring?

22 Graphical Interpretation of Average & Instantaneous Velocity

x2 − x1 vavg = t2 − t1

x1 x3 − x2 vavg = t3 − t2 x2

x3

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Instantaneous Velocity This plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity at time t is the slope of the line tangent to the curve at t.

24 Calculating Instantaneous Velocity

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Velocity & Slope

The position vs. time graph of a particle moving at constant velocity has a constant slope.

4.5 m The position vs. time graph of a particle 3.0 s moving with a changing velocity has a changing slope = velocity = 4.5 m/3.0 s = 1.5 m/s slope.

26 Key Points of Lecture 2 • Scalars and Vectors •Coordinate Systems r •Position ( x; xi, xf ) & Displacement ( ∆ x ) r •Average Speed ( vavg ) & Velocity ( v avg ) •Instantaneous Speed ( v ) and Velocity ( vr ) •Relation between velocity & slope of position-time plot z Before the next lecture, read Walker, 2.4 - 2.6. z Homework Assignment #2a is due at 11:00 PM on Wednesday, Sept. 2.

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