4 - Vectors & relative motion

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4 - Vectors & relative motion

  1. 1. VECTORS & RELATIVE MOTION 1. Show vector direction as a bearing, compass direction or angle relative to a fixed axis. 2. Perform simple vector arithmetic (including addition, subtraction) 3. Resolve vectors into components 4. Define the term relative velocity 5. Use vectors to solve relative velocity problems Reading: Chapter 8 (p91 to 103)
  2. 2. VECTOR DIRECTION Definition A vector is a physical quantity that has both size and direction. Examples Displacement, velocity, acceleration and force. A vector has a head and a tail head head tail tail A vector’s direction can be described in a number of different ways: (a)Bearings A bearing is an angle measure clockwise from the North. Eg. N N N 30o 45o 30o Direction = 045o Direction = Direction =
  3. 3. (b)Compass reference An angle is given from the N, S, E or W direction Eg. N N N 30o 45o 30o Direction = Direction = Direction = (c) Referenced from the vertical or horizontal An angle is given from a vertical or horizontal axis Eg. 20o 60o Direction = Direction =
  4. 4. VECTOR ARITHMETIC Addition Two vectors are added head to tail to produce a resultant vector. The resultant vector is a single vector that has the same effect as the two vectors combined. Eg: Adding two vectors, a and b: head head b ~ a ~ tail tail a b ~+ ~ Ex.8A Q.1 to 8
  5. 5. Subtraction Vector subtraction is “addition of the opposite”. The opposite of a vector is a vector which has the same magnitude but is opposite in direction. Eg: Subtracting two vectors, a minus b: b ~ a ~ -b ~ a b a - b ~ - ~ = ~ + ~
  6. 6. VECTOR COMPONENTS Any vector can be drawn as the sum of two other vectors which are drawn at right angles to each other. These two vectors are called components. Examples - Resolving vectors into components: 1 Any vector can be expressed as the sum of two components (a) (b) (c) 2 Horizontal and vertical components are the most useful components 30 N 30 N Vertical component 25o 25o Horizontal component The vector is the sum of its horizontal and vertical components
  7. 7. CALCULATING THE SIZE OF THE COMPONENTS 3 It is possible to calculate the component of a vector along any axis 30 N F ~ 25o f ~ This is the component of the vector, F along this axis ~ f = F.cos25o Example Calculate the size of the component of the following velocity vector along the axis shown: 50 ms-1 Exercises: “Vector arithmetic & components”
  8. 8. CHANGE IN VELOCITY The change in velocity of a moving object is the final velocity minus the initial velocity: ∆v = vf - vi Example 1. A tennis ball falls to the ground, striking at right angles. It bounces off the ground and travels along the same path on the rebound as it travelled as it was falling. The initial velocity is 10 ms-1 vertically downwards. It rebounds with a final velocity of 10 ms-1 upwards. Calculate the change in velocity of the ball. 10 ms-1 10 ms-1
  9. 9. Example 2. A billiard ball strikes the cushion of a billiard table at an angle of 20o and rebounds at the same angle. Use a vector diagram to calculate the change in velocity. 8 2 ms-1 20o 20o 2 ms-1 Vector subtraction Calculation Ex.8A Q.9 to 14
  10. 10. Relative velocity is the velocity of an object in relation to another object. This other object can be stationery (like the ground) or moving. RELATIVE VELOCITY along a straight line Consider the example of a train travelling in a straight line along a track. A boy is standing still on the roof of a train which is travelling slowly in a straight line at a speed of 2 kmh-1. 2 kmh-1 A second boy standing on the ground holding a speed gun measures the boy’s velocity at 2 kmh-1.
  11. 11. Relative velocity is the velocity of an object in relation to another object. This other object can be stationery (like the ground) or moving. RELATIVE VELOCITY along a straight line Consider the example of a train travelling in a straight line along a track. A boy is standing still on the roof of a train which is travelling slowly in a straight line at a speed of 2 kmh-1. 2k A second boy standing on the ground holding a speed gun measures the boy’s velocity at 2 kmh-1.
  12. 12. Relative velocity is the velocity of an object in relation to another object. This other object can be stationery (like the ground) or moving. RELATIVE VELOCITY along a straight line Consider the example of a train travelling in a straight line along a track. A boy is standing still on the roof of a train which is travelling slowly in a straight line at a speed of 2 kmh-1. 2k A second boy standing on the ground holding a speed gun measures the boy’s velocity at 2 kmh-1. The boy’s velocity is 2 kmh-1 relative to the ground.
  13. 13. A boy is standing still on the roof of a train which is travelling slowly in a straight line at a speed of 2 kmh-1. 2 kmh-1 A second boy standing on the roof of the train measures the boy’s velocity at 0 kmh-1.
  14. 14. A boy is standing still on the roof of a train which is travelling slowly in a straight line at a speed of 2 kmh-1. 2 A second boy standing on the roof of the train measures the boy’s velocity at 0 kmh-1.
  15. 15. A boy is standing still on the roof of a train which is travelling slowly in a straight line at a speed of 2 kmh-1. 2 A second boy standing on the roof of the train measures the boy’s velocity at 0 kmh-1. The boy’s velocity is 0 kmh-1 relative to the train.
  16. 16. TWO VELOCITIES RELATIVE TO EACH OTHER Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is travelling considerably slower (at 12 kmh-1). Car 20 kmh-1 Bus 12 kmh-1 • The velocity of the car relative to the bus is 8 kmh-1 in the forward direction because an observer in the bus sees the car moving forward (past the bus) at 8 kmh-1. • The velocity of the car relative to the bus is the velocity of the car minus the velocity of the bus. We can write this as a simple equation: vcb = vc - vb Where c = car ~ ~ ~ b = bus The vector subtraction is shown below:
  17. 17. TWO VELOCITIES RELATIVE TO EACH OTHER Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is travelling considerably slower (at 12 kmh-1). Car 20 kmh-1 Bus 12 kmh-1 • The velocity of the car relative to the bus is 8 kmh-1 in the forward direction because an observer in the bus sees the car moving forward (past the bus) at 8 kmh-1. • The velocity of the car relative to the bus is the velocity of the car minus the velocity of the bus. We can write this as a simple equation: vcb = vc - vb Where c = car ~ ~ ~ b = bus The vector subtraction is shown below: 20 kmh-1
  18. 18. TWO VELOCITIES RELATIVE TO EACH OTHER Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is travelling considerably slower (at 12 kmh-1). Car 20 kmh-1 Bus 12 kmh-1 • The velocity of the car relative to the bus is 8 kmh-1 in the forward direction because an observer in the bus sees the car moving forward (past the bus) at 8 kmh-1. • The velocity of the car relative to the bus is the velocity of the car minus the velocity of the bus. We can write this as a simple equation: vcb = vc - vb Where c = car ~ ~ ~ b = bus The vector subtraction is shown below: 12 kmh-1 20 kmh-1
  19. 19. TWO VELOCITIES RELATIVE TO EACH OTHER Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is travelling considerably slower (at 12 kmh-1). Car 20 kmh-1 Bus 12 kmh-1 • The velocity of the car relative to the bus is 8 kmh-1 in the forward direction because an observer in the bus sees the car moving forward (past the bus) at 8 kmh-1. • The velocity of the car relative to the bus is the velocity of the car minus the velocity of the bus. We can write this as a simple equation: vcb = vc - vb Where c = car ~ ~ ~ b = bus The vector subtraction is shown below: 8 kmh-1 12 kmh-1 20 kmh-1
  20. 20. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank N 2 ms-1 2 ms-1
  21. 21. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank N N 2 ms-1 2 ms-1 2 ms-1 -1 2 ms
  22. 22. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank 2 ms-1 N N N 2 ms-1 2 ms-1 2 ms-1ms-1 2 ms-1 2
  23. 23. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank 2 ms-1 2 ms-1 N N N N 2 ms-1 2 ms-1 2 ms-1ms-1 -1 2 ms-1ms 2 2
  24. 24. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank 2 ms-1 2 ms-1 2 ms-1 N N N N N 2 ms-1 2 ms-1 2 ms-1ms-1ms-1 2 ms-1ms-1 2 2 2
  25. 25. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank 2 ms-1 2 ms-1 2 ms-1 2 ms-1 N N N N N N 2 ms-1 2 ms-1 2 ms-1ms-1ms-1 -1 2 ms-1ms-1ms 2 2 2 2
  26. 26. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank 2 ms-1 2 ms-1 2 ms-1 2 ms-1 2 ms-1 N N N N N N N 2 ms-1 2 ms-1 2 ms-1ms-1ms-1ms-1 2 ms-1ms-1ms-1 2 2 2 2 2
  27. 27. RELATIVE VELOCITY in 2D Consider the example of an aircraft carrier which is floating down a canal with a speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to the sides of the vessel with a speed of 2 ms-1 as shown. N The bank 2 ms-1 2 ms-1 2 ms-1 2 ms-1 2 ms-1 N N N N N N N 2 ms-1 2 ms-1 2 ms-1ms-1ms-1ms-1 2 ms-1ms-1ms-1 2 2 2 2 2 The officer’s velocity relative to the moving object (the boat) is 2 ms-1 North The velocity of the moving object is 2 ms-1 East The officer’s velocity is a combination of these two velocities.
  28. 28. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors:
  29. 29. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object
  30. 30. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1
  31. 31. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object
  32. 32. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object 2 ms-1
  33. 33. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object 2 ms-1 2 ms-1
  34. 34. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object 2 ms-1 2 ms-1 2 ms-1
  35. 35. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object 2 ms-1 2 ms-1 2 ms-1 The officer’s velocity, v
  36. 36. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object 2 ms-1 2 ms-1 2 ms-1 v = 22 + 22 The officer’s velocity, v
  37. 37. The officer’s direction is North - East. The officer’s velocity can be calculated by adding the two velocity vectors: The officer’s velocity relative to the moving object 2 ms-1 The velocity of the moving object 2 ms-1 2 ms-1 2 ms-1 v = 22 + 22 The officer’s velocity, v = 2.8 ms-1 NE
  38. 38. Intuitively: The officer is moving at 2 ms-1 North at the same time as he is moving at 2 ms-1 East. It follows that the officer’s velocity (relative to the ground) is the sum of these 2 motions. In other words: “The velocity of the officer relative to the ground is equal to the velocity of the officer relative to the boat plus the velocity of the boat relative to the ground” We can write this as an equation: Where o = officer vog = vob + vb g b = boat ~ ~ ~ g = ground PROCESS FOR PROBLEM-SOLVING 1. Read the question carefully ---> Underline relevant information. 2. Assign a variable (a letter) to each object in the system. Eg. let g = the ground 3. Construct the vector symbol equation that relates the relative velocities to each other. 4. Rearrange the equation if required. 5. Draw the vector diagram from the equation. 6. Solve for the unknown quantity.
  39. 39. EXAMPLES 1. A plane flies East with an air speed of 500 kmh-1 (this is the velocity of the plane relative to the air). There is a wind blowing in the opposite direction at a speed of 30 kmh-1. Calculate the velocity of the plane relative to the ground by answering the questions below: (a) Give letters to symbolise the air, plane and ground. (b) Write a vector equation that relates the 3 velocities. Below each vector symbol sketch the vector showing size and direction. (c) Perform the vector calculation to get your answer.
  40. 40. 2. A plane flies at 60 ms-1 due North, relative to the ground. A 20 ms-1 Northwest wind is blowing (i.e. air coming from the Northwest). Find: (a) the airspeed of the plane ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ (b) the direction that the pilot has to aim the plane to achieve his 20 ms-1 velocity relative to the ground. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________
  41. 41. 3. A ferry crosses a 500 m wide river in 1.5 minutes. The ferry must travel across the river in a straight line from one ferry landing to the other as shown in the diagram (below). The ferry has a water speed of 10 ms-1. Find the speed of the current. Ferry landing Current 500 m Ferry Ferry landing __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ Ex.8B Q.1 to 6
  42. 42. 12 PHYSICS RELATIVE VELOCITY ASSIGNMENT Name 1. A man swims in a river which has a current of 2.5 ms-1. He finds that he can swim upstream, against the current, at 1.5 ms-1 relative to the river bank. bank river current bank (a) What is his velocity relative to the water? ________________________________________________________________ (b) He then swims downstream with the same effort. What is his velocity relative to the bank? ________________________________________________________________ ________________________________________________________________ (c) The man then swims (at 4 ms-1 relative to the water) so that he faces directly across the river. As he swims across, the current takes him downstream a little. Draw a vector velocity triangle to show this.
  43. 43. 2. A small plane is flying West at 60 ms-1. A 20 ms-1 South wind springs up. (a) Sketch this situation (b) The pilot has to continue at 60 ms-1 Westwards despite the wind. Sketch a vector triangle which shows what the pilot should do. (c) Calculate the direction of the plane’s air speed (i.e. its speed relative to the air). ________________________________________________________________ ________________________________________________________________ (d) Calculate the magnitude of the plane’s air speed. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________
  44. 44. 3. A man rows his boat North directly across a river at 2.0 ms-1. A current starts to flow East at 1.3 ms-1. bank (a) Sketch a vector triangle of the velocities. Label each vector. river current Boat bank (b) Write a vector equation for the velocities. ________________________________________________________________ (c) Calculate the magnitude of the boat’s velocity relative to the bank. ________________________________________________________________ ________________________________________________________________ (d) Calculate the direction of the boat’s velocity relative to the bank. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

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