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Kinematics in Two Dimensions

Kinematics in Two Dimensions. Chapter 3. 3.1 Displacement, Velocity, and Acceleration. 3.1 Displacement, Velocity, and Acceleration. Average velocity is the displacement divided by the elapsed time. 3.1 Displacement, Velocity, and Acceleration.

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Kinematics in Two Dimensions

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  1. Kinematics in Two Dimensions Chapter 3

  2. 3.1 Displacement, Velocity, and Acceleration

  3. 3.1 Displacement, Velocity, and Acceleration Average velocity is the displacement divided by the elapsed time.

  4. 3.1 Displacement, Velocity, and Acceleration The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

  5. 3.1 Displacement, Velocity, and Acceleration

  6. 3.1 Displacement, Velocity, and Acceleration DEFINITION OF AVERAGE ACCELERATION

  7. 3.1.1. Which one of the following statements concerning the displacement of an object is false? a) Displacement is a vector quantity that points from the initial position of an object to its final position. b) The magnitude of an object’s displacement is always equal to the distance it traveled from its initial position to its final position. c) The magnitude of an object’s displacement is the shortest distance from its initial position to its final position. d) The direction of an object’s displacement is indicated by an arrow that begins on the initial position of the object and ends on its final position. e) The length of the arrow representing an object’s displacement is proportional to its magnitude.

  8. 3.1.1. Which one of the following statements concerning the displacement of an object is false? a) Displacement is a vector quantity that points from the initial position of an object to its final position. b) The magnitude of an object’s displacement is always equal to the distance it traveled from its initial position to its final position. c) The magnitude of an object’s displacement is the shortest distance from its initial position to its final position. d) The direction of an object’s displacement is indicated by an arrow that begins on the initial position of the object and ends on its final position. e) The length of the arrow representing an object’s displacement is proportional to its magnitude.

  9. 3.1.2. At time t = 0 s, the position vector of a sailboat is r0. Later, at time t, the sailboat has a position vector r. Which of the following expressions correctly indicates the displacement of the sailboat during the time interval, tt0? a) r b) r0 c) r + r0 d) rr0 e) r0r

  10. 3.1.2. At time t = 0 s, the position vector of a sailboat is r0. Later, at time t, the sailboat has a position vector r. Which of the following expressions correctly indicates the displacement of the sailboat during the time interval, tt0? a) r b) r0 c) r + r0 d) rr0 e) r0r

  11. 3.1.3. A park ranger wanted to measure the height of a tall tree. The ranger stood 6.10 m from the base of the tree; and he observed that his line of sight made an angle of 73.5° above the horizontal as he looked at the top of the tree. What is the height of the tree? a) 5.84 m b) 8.77 m c) 11.7 m d) 17.3 m e) 20.6 m

  12. 3.1.3. A park ranger wanted to measure the height of a tall tree. The ranger stood 6.10 m from the base of the tree; and he observed that his line of sight made an angle of 73.5° above the horizontal as he looked at the top of the tree. What is the height of the tree? a) 5.84 m b) 8.77 m c) 11.7 m d) 17.3 m e) 20.6 m

  13. 3.1.4. Which one of the following quantities is an object’s displacement divided by the elapsed time of the displacement? a) average velocity b) instantaneous velocity c) average displacement d) average acceleration e) instantaneous acceleration

  14. 3.1.4. Which one of the following quantities is an object’s displacement divided by the elapsed time of the displacement? a) average velocity b) instantaneous velocity c) average displacement d) average acceleration e) instantaneous acceleration

  15. 3.1.5. Which one of the following quantities is the change in object’s velocity divided by the elapsed time as the elapsed time becomes very small? a) average velocity b) instantaneous velocity c) average displacement d) average acceleration e) instantaneous acceleration

  16. 3.1.5. Which one of the following quantities is the change in object’s velocity divided by the elapsed time as the elapsed time becomes very small? a) average velocity b) instantaneous velocity c) average displacement d) average acceleration e) instantaneous acceleration

  17. 3.1.6. How is the direction of the average acceleration determined? a) The direction of the average acceleration is the same as that of the displacement vector. b) The direction of the average acceleration is the same as that of the instantaneous velocity vector. c) The direction of the average acceleration is that of the vector subtraction of the initial velocity from the final velocity. d) The direction of the average acceleration is the same as that of the average velocity vector. e) The direction of the average acceleration is that of the vector addition of the initial velocity from the final velocity.

  18. 3.1.6. How is the direction of the average acceleration determined? a) The direction of the average acceleration is the same as that of the displacement vector. b) The direction of the average acceleration is the same as that of the instantaneous velocity vector. c) The direction of the average acceleration is that of the vector subtraction of the initial velocity from the final velocity. d) The direction of the average acceleration is the same as that of the average velocity vector. e) The direction of the average acceleration is that of the vector addition of the initial velocity from the final velocity.

  19. 3.1.7. A delivery truck leaves a warehouse and travels 3.20 km east. The truck makes a right turn and travels 2.45 km south to arrive at its destination. What is the magnitude and direction of the truck’s displacement from the warehouse? a) 4.03 km, 37.4 south of east b) 2.30 km, 52.5 south of east c) 0.75 km, 37.8 south of east d) 2.40 km, 45.0 south of east e) 5.65 km, 52.5 south of east

  20. 3.1.7. A delivery truck leaves a warehouse and travels 3.20 km east. The truck makes a right turn and travels 2.45 km south to arrive at its destination. What is the magnitude and direction of the truck’s displacement from the warehouse? a) 4.03 km, 37.4 south of east b) 2.30 km, 52.5 south of east c) 0.75 km, 37.8 south of east d) 2.40 km, 45.0 south of east e) 5.65 km, 52.5 south of east

  21. 3.1.8. While on a one-hour trip, a small boat travels 32 km north and then travels 45 km east. What is the boat's average speed for the one-hour trip? a) 39 km/h b) 55 km/h c) 77 km/h d) 89 km/h e) 96 km/h

  22. 3.1.8. While on a one-hour trip, a small boat travels 32 km north and then travels 45 km east. What is the boat's average speed for the one-hour trip? a) 39 km/h b) 55 km/h c) 77 km/h d) 89 km/h e) 96 km/h

  23. 3.1.9. While on a one-hour trip, a small boat travels 33 km north and then travels 45 km east. What is the direction of the boat's average velocity for the one-hour trip? a) 45 north of east b) 54 north of east c) 37 north of east d) 27 north of east e) due east

  24. 3.1.9. While on a one-hour trip, a small boat travels 33 km north and then travels 45 km east. What is the direction of the boat's average velocity for the one-hour trip? a) 45 north of east b) 54 north of east c) 37 north of east d) 27 north of east e) due east

  25. 3.1.1. A truck drives due south for 1.2 km in 1.5 minutes. Then, the truck turns and drives due west for 1.2 km in 1.5 minutes. Which one of the following statements is correct? a) The average speed for the two segments is the same. The average velocity for the two segments is the same. b) The average speed for the two segments is not the same. The average velocity for the two segments is the same. c) The average speed for the two segments is the same. The average velocity for the two segments is not the same. d) The average speed for the two segments is not the same. The average velocity for the two segments is not the same.

  26. 3.1.1. A truck drives due south for 1.2 km in 1.5 minutes. Then, the truck turns and drives due west for 1.2 km in 1.5 minutes. Which one of the following statements is correct? a) The average speed for the two segments is the same. The average velocity for the two segments is the same. b) The average speed for the two segments is not the same. The average velocity for the two segments is the same. c) The average speed for the two segments is the same. The average velocity for the two segments is not the same. d) The average speed for the two segments is not the same. The average velocity for the two segments is not the same.

  27. 3.1.2. A ball is rolling down one hill and up another as shown. Points A and B are at the same height. How do the velocity and acceleration change as the ball rolls from point A to point B? a) The velocity and acceleration are the same at both points. b) The velocity and the magnitude of the acceleration are the same at both points, but the direction of the acceleration is opposite at B to the direction it had at A. c) The acceleration and the magnitude of the velocity are the same at both points, but the direction of the velocity is opposite at B to the direction it had at A. d) The horizontal component of the velocity is the same at points A and B, but the vertical component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B is the same. e) The vertical component of the velocity is the same at points A and B, but the horizontal component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B has the same magnitude, but opposite direction.

  28. 3.1.2. A ball is rolling down one hill and up another as shown. Points A and B are at the same height. How do the velocity and acceleration change as the ball rolls from point A to point B? a) The velocity and acceleration are the same at both points. b) The velocity and the magnitude of the acceleration are the same at both points, but the direction of the acceleration is opposite at B to the direction it had at A. c) The acceleration and the magnitude of the velocity are the same at both points, but the direction of the velocity is opposite at B to the direction it had at A. d) The horizontal component of the velocity is the same at points A and B, but the vertical component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B is the same. e) The vertical component of the velocity is the same at points A and B, but the horizontal component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B has the same magnitude, but opposite direction.

  29. 3.2 Equations of Kinematics in Two Dimensions Equations of Kinematics

  30. 3.2 Equations of Kinematics in Two Dimensions

  31. 3.2 Equations of Kinematics in Two Dimensions

  32. 3.2 Equations of Kinematics in Two Dimensions The x part of the motion occurs exactly as it would if the y part did not occur at all, and vice versa.

  33. 3.2 Equations of Kinematics in Two Dimensions Example 1 A Moving Spacecraft In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x and vx, (b) y and vy, and (c) the final velocity of the spacecraft at time 7.0 s.

  34. 3.2 Equations of Kinematics in Two Dimensions Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables associated with each direction. 4. Verify that the information contains values for at least three of the kinematic variables. Do this for x and y. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem.

  35. 3.2 Equations of Kinematics in Two Dimensions Example 1 A Moving Spacecraft In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x and vx, (b) y and vy, and (c) the final velocity of the spacecraft at time 7.0 s.

  36. 3.2 Equations of Kinematics in Two Dimensions

  37. 3.2 Equations of Kinematics in Two Dimensions

  38. 3.2 Equations of Kinematics in Two Dimensions

  39. 3.2 Equations of Kinematics in Two Dimensions

  40. 3.2.1. In two-dimensional motion in the x-y plane, what is the relationship between the x part of the motion to the y part of the motion? a) The x part of the motion is independent of the y part of the motion. b) The y part of the motion goes as the square of the x part of the motion. c) The x part of the motion is linearly dependent on the y part of the motion. d) The x part of the motion goes as the square of the y part of the motion. e) If the y part of the motion is in the vertical direction, then x part of the motion is dependent on the y part.

  41. 3.2.1. In two-dimensional motion in the x-y plane, what is the relationship between the x part of the motion to the y part of the motion? a) The x part of the motion is independent of the y part of the motion. b) The y part of the motion goes as the square of the x part of the motion. c) The x part of the motion is linearly dependent on the y part of the motion. d) The x part of the motion goes as the square of the y part of the motion. e) If the y part of the motion is in the vertical direction, then x part of the motion is dependent on the y part.

  42. 3.2.2. Complete the following statement: In two-dimensional motion in the x-y plane, the x part of the motion and the y part of the motion are independent a) only if there is no acceleration in either direction. b) only if there is no acceleration in one of the directions. c) only if there is an acceleration in both directions. d) whether or not there is an acceleration in any direction. e) whenever the acceleration is in the y direction only.

  43. 3.2.2. Complete the following statement: In two-dimensional motion in the x-y plane, the x part of the motion and the y part of the motion are independent a) only if there is no acceleration in either direction. b) only if there is no acceleration in one of the directions. c) only if there is an acceleration in both directions. d) whether or not there is an acceleration in any direction. e) whenever the acceleration is in the y direction only.

  44. 3.2.1. An eagle takes off from a tree branch on the side of a mountain and flies due west for 225 m in 19 s. Spying a mouse on the ground to the west, the eagle dives 441 m at an angle of 65 relative to the horizontal direction for 11 s to catch the mouse. Determine the eagle’s average velocity for the thirty second interval. a) 19 m/s at 44 below the horizontal direction b) 22 m/s at 65 below the horizontal direction c) 19 m/s at 65 below the horizontal direction d) 22 m/s at 44 below the horizontal direction e) 25 m/s at 27 below the horizontal direction

  45. 3.2.1. An eagle takes off from a tree branch on the side of a mountain and flies due west for 225 m in 19 s. Spying a mouse on the ground to the west, the eagle dives 441 m at an angle of 65 relative to the horizontal direction for 11 s to catch the mouse. Determine the eagle’s average velocity for the thirty second interval. a) 19 m/s at 44 below the horizontal direction b) 22 m/s at 65 below the horizontal direction c) 19 m/s at 65 below the horizontal direction d) 22 m/s at 44 below the horizontal direction e) 25 m/s at 27 below the horizontal direction

  46. 3.2.2. A space craft is initially traveling toward Mars. As the craft approaches the planet, rockets are fired and the spacecraft temporarily stops and reorients itself. Then, at time t = 0 s, the rockets again fire causing the craft to move toward Mars with a constant acceleration. At time t, the craft’s displacement is r and its velocity v. Assuming the acceleration is constant, what would be its displacement and velocity at time 3t? a) 3r and 3v b) 4r and 2v c) 6r and 3v d) 9r and 3v e) 9r and 6v

  47. 3.2.2. A space craft is initially traveling toward Mars. As the craft approaches the planet, rockets are fired and the spacecraft temporarily stops and reorients itself. Then, at time t = 0 s, the rockets again fire causing the craft to move toward Mars with a constant acceleration. At time t, the craft’s displacement is r and its velocity v. Assuming the acceleration is constant, what would be its displacement and velocity at time 3t? a) 3r and 3v b) 4r and 2v c) 6r and 3v d) 9r and 3v e) 9r and 6v

  48. 3.2.3. Cathy and Jim have an argument about which route is the fastest route between their home at point A in the drawing and their workplace at point B. Cathy drives east and then north to work with a stop sign at the turn. Jim goes north, stops at a stop sign, and then goes northeast before reaching another stop sign, at which he makes a right turn to go east. Their cars are identical; each accelerates from rest to the maximum speed on either route of 15.6 m/s in 7.74 s. For each segment, they accelerate to the maximum speed, drive at that speed, and then decelerate at a rate of 2.5 m/s2 before each stop. Who gets to work first and what is his/her average velocity? The distances of the sides labeled “a” are 1.00 km and those labeled “b” are 6.00 km. a) They arrive at the same time with an average velocity of 12.5 m/s, 45  north of east. b) Jim arrives first with an average velocity of 14.1 m/s, 45  north of east. c) Cathy arrives first with an average velocity of 12.5 m/s, 45  north of east. d) Jim arrives first with an average velocity of 11.4 m/s, 45  north of east. e) Cathy arrives first with an average velocity of 10.8 m/s, 45  north of east.

  49. 3.2.3. Cathy and Jim have an argument about which route is the fastest route between their home at point A in the drawing and their workplace at point B. Cathy drives east and then north to work with a stop sign at the turn. Jim goes north, stops at a stop sign, and then goes northeast before reaching another stop sign, at which he makes a right turn to go east. Their cars are identical; each accelerates from rest to the maximum speed on either route of 15.6 m/s in 7.74 s. For each segment, they accelerate to the maximum speed, drive at that speed, and then decelerate at a rate of 2.5 m/s2 before each stop. Who gets to work first and what is his/her average velocity? The distances of the sides labeled “a” are 1.00 km and those labeled “b” are 6.00 km. a) They arrive at the same time with an average velocity of 12.5 m/s, 45  north of east. b) Jim arrives first with an average velocity of 14.1 m/s, 45  north of east. c) Cathy arrives first with an average velocity of 12.5 m/s, 45  north of east. d) Jim arrives first with an average velocity of 11.4 m/s, 45  north of east. e) Cathy arrives first with an average velocity of 10.8 m/s, 45  north of east.

  50. 3.3 Projectile Motion Under the influence of gravity alone, an object near the surface of the Earth will accelerate downwards at 9.80m/s2.

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