Forces and Motion in One Dimension. Chapter 3

Transcription

1 Forces and Motion in One Dimension Chapter 3

2 Constant velocity on an x-versus-t graph

3 Velocity and Position In general, the average velocity is the slope of the line segment that connects the positions at the beginning and end of the time interval Section 2.2 3

4 Instantaneous Velocity Average velocity doesn t tell us anything about details during the time interval To look at some of the details, smaller time intervals are needed The slope of the curve at the time of interest will give the instantaneous velocity at that time v Will be referred to as velocity in the text t lim 0 x t Section 2.2 4

5 Table 2-1 x-versus-t Values for Figure 2-7

6 Figure 2-7 Instantaneous velocity

7 Table 2-2 Calculating the Instantaneous Velocity at t = 1 s.

8 Velocity Example A shows a multiple exposure sketch of a rocket powered car B shows the positiontime graph C shows the velocitytime graph In this case, the speed of the car increases with time Section 2.2 8

9 Velocity of a Bicycle (Example 2.1) Need average velocity from 2.0 to 3.0 seconds Find displacement Δx Find average velocity v ave = Δx / Δt Solve Section 2.2 9

10 7. A jogger maintains a speed of 3.0 m/ s for 200 m until he encounters a stoplight, and he abruptly stops and waits 30 s for the light to change. He then resumes his exercise and maintains a speed of 3.5 m/ s for the remaining 50 m to his home. ( a) What was his average velocity for this entire time interval? ( b) What were his maximum and minimum velocities, and how do they compare with this average? 10

11 A dog runs back and forth between its two owners, who are walking toward one another. The dog starts running when the owners are 10.0 m apart. If the dog runs with a speed of 3.0 m/s, and the owners each walk with a speed of 1.3 m/s, how far has the dog traveled when the owners meet? 11

12 You drive in a straight line at 20.0 m/s for 10.0 minutes, then at 30.0 m/s for another 10.0 minutes. (a) Is your average speed 25.0 m/s, more than 25.0 m/s, less than 25.0 m/s? Explain. (b) Verify your answer to part (a) by calculating the average speed. 12

13 A father paces back and forth, producing the positionversus-time graph. Without performing a calculation, indicate whether the father s velocity is positive, negative, or zero on each of the following segments of the graph: (a) A, (b) B, (c) C, and (d) D. Calculate the numerical value of the father s velocity for the segments (e) A, (f) B, (g) C, and (h) D, and show that your results verify your answers to part (a). 13

14 60. A thief is trying to escape from a parking garage after completing a robbery, and the thief s car is speeding ( v =12 m/ s) toward the door of the parking garage. When the thief is L = 30 m from the door, a police officer flips a switch to close the garage door. The door starts at a height of 2.0 m and moves downward at 0.20 m/ s. If the thief s car is 1.4 m tall, will the thief escape? 14

15 Average Acceleration Acceleration is related to how velocity changes in time Acceleration is defined as the rate at which the velocity is changing: a ave v t Units are m/s² Section

16 Acceleration Section

17 Motion Example 1 Constant Velocity = 0 The velocity is zero On the graph, the line is at v = 0 The position is constant Generally not zero Not moving, so no change in position On the graph, a horizontal line Section 3.1

18 Motion Example 2 Constant Velocity 0 The velocity is not zero On the graph, the line is at v = constant value The position is changing steadily The same Δ x occurs each second On the graph, an upward sloping straight line The slope of the positiontime curve is equal to the value of the velocity Section 3.1

19 Motion Example 3 Constant Acceleration The acceleration is a constant On the graph, a straight horizontal line Value depends on the total force exerted on (and mass of) the object The velocity is changing On the graph, this is an upward sloping straight line The position is changing Not the same change each second On the graph, a curved line Section 3.1

20 Equations to Describe Motion with Constant Acceleration v = v o + a t v o is the velocity at some initial time t = 0 It depends on what happened prior to t = 0 x = x o + v o t + ½ a t 2 x o is the position at some initial time t = 0 v 2 = v o ² + 2 a (x f - x o ) Eliminates t from the equation Which equation to use depends on what information you are given in the problem and what you are asked to find Section 3.1

21 Constant Acceleration Equations, Summary Section 3.1

22 14. An airplane must reach a speed of 200 mi/ h to take off. If the runway is 500 m long, what is the minimum value of the acceleration that will allow the airplane to take off successfully? (1 mi =1609 m)

23 20. Your car is initially traveling at a speed of 25 m/ s. As you approach an intersection, you spot a dog in the road 30 m ahead and immediately apply your brakes. ( a) If you stop the instant before you reach the dog, what was the magnitude of your acceleration? ( b) If your velocity is positive while you are slowing to a stop, is your acceleration positive or negative?

24 Running with an initial velocity of +11 m/s a horse has an average acceleration of 1.81 m/s 2. How long does it take for the horse to decrease its velocity to +6.5 m/s?

25 When you see a traffic light turn red, you apply the brakes until you come to a stop. If your initial speed was 12 m/s, and you were heading due west, what was your average velocity during braking? Assume constant deceleration. Suppose the car in the previous problem comes to rest in 35 m. How much time does this take?