Solving Problems (continued)

Transcription

1 Solving Problems

2 Solving Problems (continued)

3 Concepts of Motion Topics: Motion diagrams Position and time Velocity Acceleration Vectors and motion Sample question: As this snowboarder moves in a graceful arc through the air, the direction of his motion, and the distance between each of his positions and the next, is constantly changing. What language should we use to describe this motion?

4 Four Types of Motion We ll Study

5 Making a Motion Diagram

6 Examples of Motion Diagrams

7 The Particle Model A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object.

8 Position and Time The position of an object is located along a coordinate system. At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0).

9 Displacement The change in the position of an object as it moves from initial The change in the position of an object as it moves from initial position x i to final position x f is its displacement x = x f x i.

10 Checking Understanding Two runners jog along a track The positions are shown at 1 s time Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

11 Answer Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

12 Checking Understanding Two runners jog along a track. The times at each position are shown. Which runner is moving faster? C. They are both moving at the same speed.

13 Answer Two runners jog along a track. The times at each position are shown. Which runner is moving faster? C. They are both moving at the same speed.

14 Speed 40 m The car moves 40 m in 1 s. Its speed is = 40 1 s m s 20 m The bike moves 20 m in 1 s. Its speed is = 20 1 s m s

15 Velocity 20 m m Bike 1 moves20min1s m s. Itsvelocity is =+20 1 s s Bike 2 moves -20 min1s Itsvelocity is = m m Bike 2 moves -20 m in 1 s. Its velocity is = s s

16 Vectors and Motion A quantity that requires both a magnitude (or size) and a direction can be represented by a vector. Graphically, we represent a vector by an arrow. The velocity of this car is 100 m/s (magnitude) to the left (direction). This boy pushes on his friend with a force of 25 N to the right.

17 Exercise Alice is sliding along a smooth icy road on her sled when she suddenly Alice is sliding along a smooth, icy road on her sled when she suddenly runs headfirst into a large, very soft snowbank that gradually brings her to a halt. Draw a motion diagram for Alice. Show and label all displacement vectors.

18 Velocity Vectors

19 Example: Velocity Vectors Jake throws a ball at a 60 angle, measured from the horizontal. The ball is caught by Jim. Draw a motion diagram of the ball with velocity vectors.

20 Representations Motion diagram (student walking to school) Table of data Graph Slide 2-9

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22 Checking Understanding Here is a motion diagram of a car moving along a straight stretch of road: Which of the following velocity-versus-time graphs matches this motion diagram? A. B. C. D. Slide 2-13

23 Answer Here is a motion diagram of a car moving along a straight stretch of road: Which of the following velocity-versus-time graphs matches this motion diagram? C. Slide 2-14

24 Checking Understanding A graph of position versus time for a basketball player moving down the court appears like so: Which of the following velocity graphs matches the above position graph? A. B. C. D. Slide 2-15

25 Answer A graph of position versus time for a basketball player moving down the court appears like so: Which of the following velocity graphs matches the above position graph? C. Slide 2-16

26 Checking Understanding A graph of velocity versus time for a hockey puck shot into a goal appears like so: Which of the following position graphs matches the above velocity graph? A. B. C. D. Slide 2-17

27 Answer A graph of velocity versus time for a hockey puck shot into a goal appears like so: Which of the following position graphs matches the above velocity graph? D. Slide 2-18

28 Examples A soccer player is 15 m from her opponent s goal. She kicks the ball hard; after 0.50 s, it flies past a defender who stands 5 m away, and continues toward the goal. How much time does the goalie have to move into position to block the kick from the moment the ball leaves the kicker s foot? Cleveland l and Chicago are 340 miles apart by train. Train A leaves Cleveland going west to Chicago at 1:00 PM, traveling at 60 mph. Train B leaves Chicago going east to Cleveland at 2:00 PM, going 45 mph. At what time do the two trains meet? How far are they from Chicago at this time? Slide 2-19

29 Acceleration Acceleration is: The rate of change of velocity The slope of a velocityversus-time graph Slide 2-20

30 Checking Understanding These four motion diagrams show the motion of a particle along the x-axis. Rank these motion diagrams by the magnitude of the acceleration. There may be ties. Slide 2-21

31 Checking Understanding These four motion diagrams show the motion of a particle along the x-axis. Which motion diagrams correspond to a positive acceleration? Which motion diagrams correspond to a negative acceleration? Slide 2-22

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33 Important equations

34 Dinner at a Distance, Part I Chameleons catch insects with their tongues, which they can extend to great lengths at great speeds. A chameleon is aiming for an insect at a distance of 18 cm. The insect will sense the attack and move away 50 ms after it begins. In the first 50 ms, the chameleon s tongue accelerates at 250 m/s 2 for 20 ms, then travels at constant speed for the remaining 30 ms. Does its tongue reach the 18 cm extension needed to catch the insect during this time? Slide 2-24

35 Dinner at a Distance, Part II Cheetahs can run at incredible speeds, but they can t keep up these speeds for long. Suppose a cheetah has spotted a gazelle. In five long strides, the cheetah has reached its top speed of 27 m/s. At this instant, the gazelle, at a distance of 140 m from the running cheetah, notices the danger and heads directly away. The gazelle accelerates at 7.0 m/s² for 3.0 s, then continues running at a constant speed that is much less than the cheetah s speed. But the cheetah can only keep running for 15 s before it must break off the chase. Does the cheetah catch the gazelle, or does the gazelle escape? Slide 2-25

36 Free Fall Slide 2-26

37 Checking Understanding An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. At which point of the trajectory is the arrow s acceleration the greatest? The least? Ignore air resistance; the only force acting is gravity. Slide 2-27

38 Checking Understanding An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity. Slide 2-28

39 Answer An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity. Slide 2-29

40 The figure below shows five arrows with differing masses that were launched straight up with the noted speeds. Rank the arrows, from greatest to least, on the basis of the maximum height the arrows reach. Ignore air resistance; the only force acting is gravity. Slide 2-30

41 Examples Spud Webb, height 5'7" 57, was one of the shortest basketball players to play in the NBA. But he had an impressive vertical leap; he was reputedly able to jump 110 cm off the ground. To jump this high, with what speed would he leave the ground? A football is punted straight up into the air; it hits the ground 5.2 s later. What was the greatest t height ht reached by the ball? With what speed did it leave the kicker s foot? Passengers on The Giant Drop, a free-fall ride at Six Flags Great America, sit in cars that are raised to the top of a tower. The cars are then released for 2.6 s of free fall. How fast are the passengers moving at the end of this speeding up phase of the ride? If the cars in which they ride then come to rest in a time of 1.0 s, what is acceleration (magnitude and direction) of this slowing down phase of the ride? Given these numbers, what is the minimum possible height of the tower? Slide 2-31

42 Additional Examples When you stop a car on icy pavement, the acceleration of your car is approximately 1.0 m/s². If you are driving on icy pavement at 30 m/s (about 65 mph) and hit your brakes, how much distance will your car travel before coming to rest? As we will see in a future chapter, the time for a car to come to rest in a collision i is always about 0.1 s. Ideally, the front of the car will crumple as this happens, with the passenger compartment staying intact. If a car is moving at 15 m/s and hits a fixed obstacle, coming to rest in 0.10 s, what is the acceleration? How much does the front of the car crumple during the collision? Slide 2-38