Welcome Back to Physics 215 General Physics I

Welcome Back to Physics 215! (General Physics I – Honors & Majors) Physics 215 -Fall 2014 Lecture 01 -2 1

Current homework assignment • HW 1: – Ch. 1 (Knight): 42, 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 Lecture 01 -2 2

Workshops Two sections! – Wednesdays: 8: 25 -9: 20 AM and 2: 35 -3: 30 PM – Fridays: 8: 25 -9: 20 AM and 10: 35 -11: 30 AM – Both meet in room 208 – Attend either section each day – Don’t need to change registration Lecture 01 -2 3

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

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

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 car’s speedometer and my direction

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

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 car’s speedometer and my direction

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

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

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

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) Lecture 01 -2 12

When does vav = vinst ? • When x(t) curve is a straight line – Tangent to curve is same at all points in time x t • We say that such a motion is a constant velocity motion – we’ll see that this occurs when no forces act Lecture 01 -2 13

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

Summary of terms • Positions: xinitial, xfinal • Displacements: Dx = xfinal - xinitial • Instants of time: tinitial, tfinal • Time intervals: Dt = tfinal - tinitial • Average velocity: vav = Dx/Dt • Instantaneous velocity: v = dx/dt • Instantaneous speed: |v| = |dx/dt| Lecture 01 -2 15

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

Sample problem • A car’s 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. Lecture 01 -2 17

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

Fan cart demo • Attach fan to cart - provides a constant force (we’ll 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 Lecture 01 -2 19

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

You are throwing a ball up in the air. At its highest point, the ball’s 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 Lecture 01 -2 21

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

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! Lecture 01 -2 23

a b x T t 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<T 4. Somewhere, both trains have same a Lecture 01 -2 24

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 x – Positive curvature • a>0 – No curvature • a=0 Lecture 01 -2 25 t

Sample problem • An object’s 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. 00 s and t = 3. 00 s. – Calculate the instantaneous accel. at i) t = 2. 00 s; ii) t = 3. 00 s. Lecture 01 -2 26

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

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

Displacement – integral of velocity lim. Dt 0 S Dt v(t) = area under v(t) curve note: `area’ can be positive or negative *Consider v(t) curve for cart in different situations… v *Net displacement? t

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 a(t) T 1 Dt T 2 t

Summary • velocity v = dx/dt = slope of x(t) curve – NOT x/t !! • displacement Dx 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. Dv is ∫a(t)dt = area under a(t) curve – NOT at !!

Reading assignment • Kinematics, constant acceleration • 2. 4 – 2. 7 in textbook Lecture 01 -2 32