Slide 1

Physics 215 -Fall 2014Lecture01-21 Welcome Back to Physics 215! (General Physics I Honors & Majors)

Slide 2

Lecture 01-2 2 Current homework assignment HW1: Ch.1 (Knight): 42, 52, 56; Ch.2 (Knight): 26, 30, 40 due Wednesday Sept 3 rd in recitation TA will grade each HW set with a score from 0 to 5

Slide 3

Lecture 01-2 3 Workshops Two sections! Wednesdays: 8:25-9:20AM and 2:35-3:30PM Fridays: 8:25-9:20AM and 10:35-11:30AM Both meet in room 208 Attend either section each day Dont need to change registration

Slide 4

Velocity Definition: Average velocity in some time interval t is given by v av = (x 2 - x 1 )/(t 2 - t 1 ) = x/ t Displacement x can be positive or negative so can velocity it is a vector, too Average speed is not a vector, just (distance traveled)/ t

Slide 5

Discussion Average velocity is that quantity which when multiplied by a time interval yields the net displacement For example, driving from Syracuse Ithaca

Slide 6

Instantaneous velocity But there is another type of velocity which is useful instantaneous velocity Measures how fast my position (displacement) is changing at some instant of time Example -- nothing more than the reading on my cars speedometer and my direction

Slide 7

Describing motion Average velocity (for a time interval): v average = Instantaneous velocity (at an instant in time) v instant = v = Instantaneous speed |v|

Slide 8

Instantaneous velocity But there is another type of velocity which is useful instantaneous velocity Measures how fast my position (displacement) is changing at some instant of time Example -- nothing more than the reading on my cars speedometer and my direction

Slide 9

Describing motion Average velocity (for a time interval): v average = Instantaneous velocity (at an instant in time) v instant = v = Instantaneous speed |v|

Slide 10

Instantaneous velocity Velocity at a single instant of time Tells how fast the position (vector) is changing at some instant in time Note while x and t approach zero, their ratio is finite! Subject of calculus was invented precisely to describe this limit derivative of x with respect to t

Slide 11

Lecture 01-2 11 Velocity from graph P Q xx tt V av = x/ t As t gets small, Q approaches P and v dx/dt = slope of tangent at P t x instantaneous velocity

Slide 12

Lecture 01-2 12 Cart Demo Transmitter sends out signal which is reflected back by cart - can calculate distance to cart at any instant -- x(t) Cart is not subject to any forces on track - expect constant velocity Computer shows position vs. time plot for motion (and velocity plot)

Slide 13

Lecture 01-2 13 When does v av = v inst ? When x(t) curve is a straight line Tangent to curve is same at all points in time We say that such a motion is a constant velocity motion well see that this occurs when no forces act x t

Slide 14

Lecture 01-2 14 Interpretation Slope of x(t) curve reveals v inst (= v) Steep slope = large velocity Upwards slope from left to right = positive velocity Average velocity = instantaneous velocity only for motions where velocity is constant t x

Slide 15

Lecture 01-2 15 Positions: x initial, x final Displacements: x = x final - x initial Instants of time:t initial, t final Time intervals: t = t final - t initial Average velocity:v av = x/ t Instantaneous velocity:v = dx/dt Instantaneous speed:|v| = |dx/dt| Summary of terms

Slide 16

Lecture 01-2 16 Acceleration Similarly when velocity changes it is useful (crucial!) to introduce acceleration a a av = v/ t = (v F - v I )/ t Average acceleration -- keep time interval t non-zero Instantaneous acceleration a inst = lim t 0 v/ t = dv/dt

Slide 17

Lecture 01-2 17 Sample problem A cars velocity as a function of time is given by v(t) = (3.00 m/s) + (0.100 m/s 3 ) t 2. Calculate the avg. accel. for the time interval t = 0 to t = 5.00 s. Calculate the instantaneous acceleration for i) t = 0; ii) t = 5.00 s.

Slide 18

Lecture 01-2 18 Acceleration from graph of v(t) t v P Q T What is a av for PQ ? QR ? RT ? R Slope measures acceleration Positive a means v is increasing Negative a means v decreasing

Slide 19

Lecture 01-2 19 Fan cart demo Attach fan to cart - provides a constant force (well see later that this implies constant acceleration) Depending on orientation, force acts to speed up or slow down initial motion Sketch graphs of position, velocity, and acceleration for cart that speeds up

Slide 20

Interpreting x(t) and v(t) graphs Slope at any instant in x(t) graph gives instantaneous velocity Slope at any instant in v(t) graph gives instantaneous acceleration What else can we learn from an x(t) graph? x t v t

Slide 21

Lecture 01-2 21 You are throwing a ball up in the air. At its highest point, the balls 1.Velocity v and acceleration a are zero 2.v is non-zero but a is zero 3.Acceleration is non-zero but v is zero 4.v and a are both non-zero

Slide 22

Lecture 01-2 22 Cart on incline demo Raise one end of track so that gravity provides constant acceleration down incline (well study this in much more detail soon) Give cart initial velocity directed up the incline Sketch graphs of position, velocity, and acceleration for cart

Slide 23

Lecture 01-2 23 Acceleration from x(t) plot ? If x(t) plot is linear zero acceleration Is x(t) is curved acceleration is non-zero If slope is decreasing a is negative If slope is increasing a is positive If slope is constant a = 0 Acceleration is rate of change of slope!

Slide 24

Lecture 01-2 24 t x b a The graph shows 2 trains running on parallel tracks. Which is true: 1.At time T both trains have same v 2.Both trains speed up all time 3.Both trains have same v for some t

Lecture 01-2 25 Acceleration from x(t) Rate of change of slope in x(t) plot equivalent to curvature of x(t) plot Mathematically, we can write this as a = Negative curvature a < 0 Positive curvature a > 0 No curvature a = 0 x t

Slide 26

Lecture 01-2 26 Sample problem An objects position as a function of time is given by x(t) = (3.00 m) - (2.00 m/s) t + (3.00 m/s 2 ) t 2. Calculate the avg. accel. between t = 2.00s and t = 3.00 s. Calculate the instantaneous accel. at i) t = 2.00 s; ii) t = 3.00 s.

Slide 27

Displacement from velocity curve? Suppose we know v(t) (say as graph), can we learn anything about x(t) ? Consider a small time interval t v = x/ t x = v t So, total displacement is the sum of all these small displacements x x = x = lim t 0 v(t) t = v t

Slide 28

Graphical interpretation v t TT T2T2 tt v(t) Displacement between T 1 and T 2 is area under v(t) curve

Slide 29

Displacement integral of velocity lim t 0 t v(t) = area under v(t) curve note: `area can be positive or negative *Consider v(t) curve for cart in different situations v t *Net displacement?

Slide 30

Velocity from acceleration curve Similarly, change in velocity in some time interval is just area enclosed between curve a(t) and t-axis in that interval. a t TT T2T2 tt a(t)

Slide 31

Summary velocity v = dx/dt = slope of x(t) curve NOT x/t !! displacement x is v(t)dt = area under v(t) curve NOT vt !! accel. a = dv/dt = slope of v(t) curve NOT v/t !! change in vel. v is a(t)dt = area under a(t) curve NOT at !!

Slide 32

Lecture 01-2 32 Reading assignment Kinematics, constant acceleration 2.4 2.7 in textbook