Physic 231 Lecture 3. Main points of today s lecture. for constant acceleration: a = a; assuming also t0. v = lim

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1 Physic 231 Lecture 3 Main points of today s lecture Δx v = ; Δ t = t t0 for constant acceleration: a = a; assuming also t0 = 0 Δ x = v v= v0 + at Δx 1 v = lim Δ x = Δ t 0 ( v+ vo ) t 2 Δv 1 2 a = ; Δ v= a Δ x = v0t+ at Δv v v0 = 2aΔx a = limδ t 0

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3 Displacement Displacement = difference between final position and initial position Δ! r =! r! r 0 Δx = x x 0 (x x(t), x 0 x(t 0 )) Displacement is a vector, just like position; can be negative Displacement is independent of choice of origin of coordinate system (unlike position) Displacement for going from point b to point a is exactly the negative of going from point a to point b: Average velocity is displacement over the time for the displacement. definition: v = Δx ; 32

4 Distance The distance is the absolute value of the displacement Distance is always positive (or zero) Distance is a scalar distance traveled =Δ x = x x 0 If the displacement is not in a straight line or is not all in the same direction, then the displacement must be broken up into segments that are straight and unidirectional to calculate the distance Average speed = distance divided by the time to go that distance speed is a scalar: average speed = Δx ; 33

5 x (mi) Suppose the man walks at a rate of 3 mi/hr from Park Lane to Hagadorn and then slows to 1.5 mi/hr from Hagadorn to Collingwood. Then what is his average velocity? a) -2 mi/hr b) 2 mi/hr c) mi/hr d) 2.25 mi/hr Use definition: v = Δx ; Δx=Δx 1 + Δx 2 = -1+(-1) mi=-2mi = ; = 1 3 h h = 1h 1 = Δx 1 v 1 = 1mi 3mi / h = 1 3 h; 2 = Δx 2 v 2 = v = 2mi 1h = 2mi / h 1mi 1.5mi / h = 2 3 h ( ) = 1 2 Different guess: v = 1 2 v 1 + v 2 ( 3+ ( 1.5))mi / h = 2.25mi / h Wrong. You can average over time this way, but not distance. Using definition: v = Δx = v v 2 2 = v v 2 2 v = 3mi / h ( 1.5) 2 3 = 2mi / h

6 Different averages George trains for a race by breaking a 6000 m long run into three 2000 meter long sections that are all in a straight line in the positive x direction. He runs the first section at 6 mi/h. His second is at 10 mi/h. His third section is at 14 mi/h. What is his average velocity? Δx=Δx 1 + Δx 2 + Δx 3 = 3Δx 1 = ; 1 = Δx 1 v 1 ; 2 = Δx 2 v 2 = Δx 1 v 2 ; 3 = Δx 3 v 3 = Δx 1 v 3 v = Δx = 3Δx 1 Δx 1 + Δx 1 + Δx = 1 v 1 v 2 v v 1 v 2 v 3 Sam trains for the same race by running for 300s at 6 mi/h. He switches to 10 mi/h for the next 300s. Then he continues for the next 300s at 14 mi/h. What is his average velocity? Harmonic average time average: v = Δx = v v v 3 3 = v v v 3 3 v = v 1 + v 2 + v 3 3 Arithmetic average

7 Reading quiz The slope at a point on a position vs. time graph of an object is a) the object s speed at that point. b) the object s average velocity at that point. c) the object s instantaneous velocity at that point. d) the object s acceleration at that point. e) the distance traveled by the object at that point. The answer is c.

8 Graphical determination of velocity 1 To determine velocity at any point, just compute the slope. v 1 = 400m 200s = 2m / s v 2 = 0m 400s = 0m / s v 3 = 400m 400s = 1m / s v tot = x f x i = 400m 0 tot 1800s = 0.222m / s average speed = d 1 + d 2 tot = 1200m + 800m 1800s = 1.11m / s

9 Graphical determination of velocity 2 To determine instantaneous velocity at any point, just compute the slope of tangent line. You can do this by making ever smaller triangles: v = limδ t 0 Δx = + 26m 5s = 5.2m / s

10 Clicker question Hint: Which mass moves with constant velocity? x = vt Masses P and Q move with the position graphs shown. Do P and Q ever have the same velocity? If so, at what time or times? a) P and Q have the same velocity at 2 s. b) P and Q have the same velocity at 1 s and 3 s. c) P and Q have the same velocity at 1 s, 2 s, and 3 s. d) P and Q never have the same velocity. The answer is a).

11 Question 2.6 Graphing Velocity II The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? a) it speeds up all the time b) it slows down all the time c) it moves at constant velocity d) sometimes it speeds up and sometimes it slows down e) not really sure x t

12 Question 2.6 Graphing Velocity II The graph of position vs. time for a car is given below. What can you say about the velocity of the car over time? a) it speeds up all the time b) it slows down all the time c) it moves at constant velocity d) sometimes it speeds up and sometimes it slows down e) not really sure The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity! At large t, the value of the position x does not change, indicating that the car must be at rest. x t

13 Exercise A particle moves with the position vs. time graph shown. Which graph best illustrates the velocity of the particle as a function of time? The answer is A.

14 Acceleration The average acceleration is defined by the change of velocity with time: a = Δv ; Δv = v v 0 In analogy, the instantaneous acceleration can be computed using the same equation, but in the limit of very small elapsed times. a Δv lim Δ 0 = t Example: A speed boat starts from rest and reaches 3.2 m/s in 2s. What is its average acceleration? Assuming its acceleration to be constant, what is its velocity after 5 s?