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

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

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 –Don’t need to change registration

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

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

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 car’s speedometer and my direction

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

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 car’s speedometer and my direction

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

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

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

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)

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 –we’ll see that this occurs when no forces act x t

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

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

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

17. Lecture 01-2 17 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.

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

19. Lecture 01-2 19 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

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

21. Lecture 01-2 21 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

22. Lecture 01-2 22 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

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!

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<T 4.Somewhere, both trains have same a T

25. 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

26. Lecture 01-2 26 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 = 2.00s and t = 3.00 s. –Calculate the instantaneous accel. at i) t = 2.00 s; ii) t = 3.00 s.

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

28. Graphical interpretation v t TT T2T2 tt v(t) Displacement between T 1 and T 2 is area under v(t) curve

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?

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 TT T2T2 tt a(t)

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 !!

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