PHYSICS 149: Lecture 8

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1 PHYSICS 149: Lecture 8 Chapter Position and Displacement 3.2 Velocity and Acceleration Lecture 8 Purdue University, Physics 149 1

2 ILQ 1 θ N T Find W2? NOTE: because of ideal pulley and cord W2 = T A) W sinθ W B) W cosθ C) N sinθ W2 D) N cosθ Lecture 8 Purdue University, Physics 149 2

3 Motion in One Dimension -x 0 The variables are time and distance t = 0 start of observations at a point x 0 t = t end of the observations at a point x f +x Objects are in motion and velocity is (change in distance)/time Velocity can change => acceleration (change in velocity)/time All quantities except time are vectors but the vector nature is contained in whether the quantity is positive or negative Lecture 8 Purdue University, Physics 149 3

4 Position Vector To describe position, we need a reference point (origin), a distance from the origin, and a direction from the origin. object at (x,y) Position Vector (or Position) A vector quantity that t consists of the distance and direction An arrow starting at the origin and ending with the arrowhead on the object Position vector is usually denoted by r. The x-, y-, and z- component of r are usually written simply as x, y, and z (instead of r x, r y, and r z ). Lecture 8 Purdue University, Physics 149 4

5 Position A vector quantity describing where you are relative to an origin Point A is located at x=3, y=1 or (3,1) Point B is located at (-1,-2) The vector r A indicating the position of A starts at tthe origin i and dterminates t with arrowhead A Same for r B and B -3 Lecture 8 Purdue University, Physics B y 3-3 A 3 x

6 Distance vs. Displacement Distance (scalar) Total length of path traveled The path of an object does matter Displacement (vector) The change of the position vector ( r), that is, the final position vector (r f ) minus the initial position vector (r i ) = r f + ( r i ) An arrow starting at the initial position (the tip of the initial position vector) and ending with the arrowhead at the final position (the tip of the final position vector) The path of an object does not matter. The displacement depends d only on the starting ti and ending points. Lecture 8 Purdue University, Physics 149 6

7 Displacement (m) A vector quantity describing a change in position r = r f - r i The displacement from A to B is We can determine the components x-direction: x f -x i = -1 3 = -4 y-direction: y f -y i = -2 1 = -3 r = (-4, -3) r = sqrt( ) = 5 NOTE: The displacement is not the distance traveled -3 B y 3-3 A 3 x Lecture 8 Purdue University, Physics 149 7

8 Displacement Given the train s initial position and its final position what is the total t displacement of the train? A) -26 km D) -23 km B) -29 km C) +29 km E) +23 km Lecture 8 Purdue University, Physics 149 8

9 Vector Displacement Given that an object starts at A and arrives at B what is it s displacement? Lecture 8 Purdue University, Physics 149 9

10 ILQ 2 Ben leaves his home and walks to the bank, then to the park. What is the distance that Ben has walked, and, what is his displacement? A) 3 blocks, 3 blocks east, respectively B) 3 blocks, 7 blocks east, respectively C) 7 blocks, 3 blocks east, respectively D) 7 blocks, 7 blocks east, respectively Lecture 8 Purdue University, Physics

11 ILQ: Displacement 1D You travel 60 miles north (toward Chicago), then turn around and drive 10 miles back towards WL. What is the magnitude of your net displacement? A) 50 miles B) 60 miles C) 70 miles 60 miles N 50 miles N 10 miles S Net displacement is 50 miles north! Note: this is different from the distance traveled! Lecture 8 Purdue University, Physics

12 ILQ: Displacement 2D You travel 40 miles north (toward Chicago), then turn east and drive 30 miles east. What is the magnitude of your net tdisplacement? A) 10 miles B) 50 miles C) 70 miles ( ) miles + (40miles) = 2500miles = 50miles Net displacement is 50 miles NE, again this is different from distance traveled! N miles miles E Lecture 8 Purdue University, Physics

13 ILQ: Displacement and Distance You and your dog go for a walk to a nearby park. On the way, your dog takes many short side trips to chase squirrels, examine fire hydrants, and so on. When you arrive at the park, do you and your dog have the same displacement? Have you traveled the same distance? A) Same distance and same displacement B) Different displacement, same distance C) Different displacement, different distance D) Same displacement, different distance Lecture 8 Purdue University, Physics

14 Velocity (m/s) The average velocity is the change in position (vector) divided by the change in time. x(t) v Δr x x = = f Δt t t f 0 0 Instantaneous velocity is the limit of average velocity as t gets small. It is the slope of the x(t) plot. v = lim Δ t 0 Δ r Δt x(t) Lecture 8 Purdue University, Physics Δt Δx t t

15 Average Speed vs. Average Velocity Average Speed (scalar) average speed = distance elapsed time Average speed is a measure of how fast an object can actually travel during a time interval. Average Velocity (vector) displacement = elapsed time Average velocity is a measure of how fast, and in what direction, an object can move during a time interval. The x- and y-components of the average velocity are: Typical units for speed and velocity s magnitude are m/s, km/s, miles/hr (mph), and so on. Lecture 8 Purdue University, Physics

16 Velocity The following plots are x vs t x x x x t t t t Which plot represents an object at rest? Which plot represents an object with a uniform velocity in the -x direction? Lecture 8 Purdue University, Physics

17 Velocity Find the velocity for the object during the period t=5 and t=6 seconds x x 1m 6m 5m v= f o = = = 5m/sec t t 6sec 5sec 1sec f o Lecture 8 Purdue University, Physics

18 ILQ 3 Ben leaves his home and walks to the bank, then to the park. It takes 1 hour. What is the average speed of Ben and what is his average velocity? A) 3 blocks/hour, 3 blocks/hour east, respectively B) 3 blocks/hour, 7 blocks/hour east, respectively C) 7 blocks/hour, 3 blocks/hour east, respectively D) 7 blocks/hour, 7 blocks/hour east, respectively Lecture 8 Purdue University, Physics

19 Instantaneous Velocity Instantaneous velocity is the average velocity we measure when the time interval is infinitesimally it i short. The direction of the instantaneous velocity is the direction of motion and tangent to the actual path through space. The x- and y-components of the instantaneous velocity are: Lecture 8 Purdue University, Physics

20 Graph of Position x vs. Time t Graph of A vs. B = Graph of A(B) A for vertical axis and B for horizontal axis Lecture 8 Purdue University, Physics

21 ILQ The graph of position vs. time is given for a train. What can you say about the train's motion from time t 1 to time t 2? a) It is moving with constant velocity. b) It is not moving. c) It is moving with a constant negative velocity. d) It is accelerating. Lecture 8 Purdue University, Physics

22 ILQ: Average Velocity A stone is thrown upward and reaches a height y. Consider an elapsed time t, measured from the time the stone was first thrown to the time the stone has fallen back down to the ground. The magnitude of the average velocity of the stone during this time is: A) 0 C) y/t B) 2y/t D) y/2t Displacement = 0 Lecture 8 Purdue University, Physics

23 ILQ: Velocity If an object is traveling at a constant velocity, is it necessarily traveling in a straight line? A) Yes B) No Velocity is a vector!!! Lecture 8 Purdue University, Physics

24 ILQ: Velocity If the average velocity of a car during a trip along a straight road is positive, is it possible for the instantaneous velocity at some time during the trip to be negative? A) Yes B) No The car might have reversed along the trip creating a negative instantaneous velocity at the point. If the overall displacement of the car is positive for that interval than the average velocity is positive as well. x(t) B C A D v ΔAD = Δtt Lecture 8 Purdue University, Physics t

25 How determine v from x=x(t) v is the slope of the curve x versus t. A) Positive Zero Negative B) Positive Zero Negative C) Positive Zero Negative D) Positive Zero Negative x(t) A B C D t Lecture 8 Purdue University, Physics

26 How to determine v from v(x) Lets us determine the instantaneous velocity at t=0 and t=1 t=0, v slope at t=0 v m = = 1 s t=1, v slope at t=1 v 5 2 m = 2 s 2 m/ s m/ s Lecture 8 Purdue University, Physics

27 How to determine x from v(t) The total displacement x during and time t is the area under the graph v(t) Assume 1D, and v constant: v v x Δx = v = v 1x av, x Δ t Δx Δ x= v Δt 1x t 1 t 2 t Lecture 8 Purdue University, Physics

28 ILQ A jogger is exercising along a straight road that runs north-south. She starts out heading north. What is the displacement of the jogger for the entire 30.0 min? A) 2400 m, north B) 3840 m, north C) 2400 m, south D) 3120 m, south A = (1/2)(1m/s)(120s) + (1m/s)(120s) + (1/2)(2m/s)(240s) + (1m/s)(240s) + (3m/s)(120s) + (1/2)(1m/s)(120s) + (3m/s)(120s) + (4m/s)(360s) - (2m/s)(360s) + (1/2)(2m/s)(240s) = 2400m Lecture 8 Purdue University, Physics