Distance vs. Displacement

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1 Distance vs. Displacement Assume a basketball player moves from one end of the court to the other and back. Distance is twice the length of the court Distance is always positive Displacement is zero x = x f x i = 0 since x f = x i How far does the earth travel in one year? In terms of distance, quite far (the circumference of the earth's orbit is nearly one trillion meters), but in terms of displacement, not far at all (zero, actually). At the end of a year's time the earth is right back where it started from. It hasn't gone anywhere. 1

2 It takes the racing car 10 minutes(0.17 hours) to travel 23km, from start to finish. Why is the velocity lower than the speed?? Velocity can have positive and negative values. Speed can only be positive. Distance is always positive, however Displacement can have negative or positive values Why? Since the velocity has negative and positive values some will cancel each other out 2

3 = 3

Clicker Quiz You drive to the grocery You have 30 seconds to provide an answer store which is located conveniently a straight line path from your

4 Clicker Quiz You drive to the grocery You have 30 seconds to provide an answer store which is located conveniently a straight line path from your house. Does the odometer in your car measure distance or displacement from your house to the grocery store? a) distance b) displacement c) both 4

5 = 5

6 Position as a function of time (1.25,25) (1.5,24) (0.75,22) (0.5,17) (2,15) What is the average speed? (0,0) 6

7 Position as a function of time = This is the same plot of position versus time shown on Friday. The triangles represent the average velocity between t 2 and t 3 and t 4 and t 5. We want to determine the velocity nearly at the instant of t 2 and t 4. To do this we must bring t 3 closer to t 2 and t 5 closer t 4 7

8 Position as a function of time = 8

9 Position as a function of time = 9

10 Position as a function of time = 10

11 Position as a function of time = 11

12 We want to know the average velocity very close to t 1 If we let Δt=0, average velocity will approach the instantaneous velocity at t 1 Although this may appear to be a mathematical disaster, generally most continuous functions converge to finite number. As an example let s consider the quadratic function Instantaneous velocity for the quadratic position function Δt=0 12

13 Instantaneous velocity for the quadratic position function Note if α = 0, the average velocity and instantaneous velocity are both constant and identical, independent of the delta time or time. Since the velocity changes linearly, can write the following Velocity at t=t 1 Velocity at t=t 2 Velocity at midpoint time between t 1 and t 2 Average velocity is the average of the initial and final velocity for a quadratic position function 13

14 Clicker Quiz 1) When is the average velocity of an object equal to the instantaneous velocity? A) always B) never C) only when the velocity is constant D) only when the velocity is increasing at a constant rate E) only when the velocity is decreasing at a constant rate 14

15 Velocity as a function of time (1.5,24) Acceleration (0.5,17) (2,15) Acceleration (0,0) 15

16 Acceleration Just as velocity is the rate of change of displacement with time, acceleration is the rate of change of velocity with time. A particle accelerates whenever velocity changes It accelerates when velocity increases or decreases. Average acceleration: 16

17 Let s revisit the quadratic equation as the relationship between time and position Initial position at t=0 Initial velocity at t=0 Constant acceleration 17

18 Position as a function of Time dx dt Let s assume we measured an object s position as a function of time and acquired the following correlation. In this case it turns out the function t squared nicely reproduces the data. Now we may ask the question what is the instantaneous velocity of the object at every point. If we calculate the slope at every point on this curve, we would then have the following. 18

19 Position Slope as a function of Time Now we have the relationship between the position slope as function of time. In other words this is the time rate change of position versus time which is the velocity as a function of time. For our measurement, the velocity time dependence can be represented by a straight line. Since the slope of this relationship is non zero, the velocity is changing with time at a constant rate. Now if we consider the slope of this velocity relationship, we see that it is constant. Nevertheless we calculate the slope for all points and we get the following. 19

20 Velocity Slope as a function of Time As you see the velocity slope is the same for all points. Consequently the time rate change in velocity is constant so we can say acceleration is a constant. 20

Chapter 2. Motion along a straight line

Chapter 2. Motion along a straight line Chapter 2 Motion along a straight line Motion We find moving objects all around us. The study of motion is called kinematics. Examples: The Earth orbits around the Sun A roadway moves with Earth s rotation