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# Chapter 8

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### Chapter 8

1. 1. Chapter 8 – Mapping Space and Time <ul><li>8.2 – Maps and Vectors </li></ul><ul><li>To review last lesson, try these: </li></ul><ul><li>What can be found out from a distance-time graph? </li></ul><ul><li>What is the equation for Speed? </li></ul><ul><li>Why is Velocity different to Speed? </li></ul><ul><li>What is the area under a Speed-Time graph? </li></ul>
2. 2. <ul><li>Answers </li></ul><ul><li>What can be found out from a distance-time graph? </li></ul><ul><li>Instantaneous and Average Speed </li></ul><ul><li>What is the equation for Speed? </li></ul><ul><li>Distance / Time </li></ul><ul><li>Why is Velocity different to Speed? </li></ul><ul><li>It has direction </li></ul><ul><li>What is the area under a Speed-Time graph? </li></ul><ul><li>Total distance travelled </li></ul>
3. 3. <ul><li>What do Vectors tell us? </li></ul><ul><li>Vectors have: </li></ul><ul><ul><li>Magnitude </li></ul></ul><ul><ul><li>Direction </li></ul></ul>But what about a journey which has more than one direction? 50 Km in this direction 70 Km in this direction Could you get there another way? A B
4. 4. <ul><li>What do Vectors tell us? </li></ul><ul><li>Vectors have: </li></ul><ul><ul><li>Magnitude </li></ul></ul><ul><ul><li>Direction </li></ul></ul>But what about a journey which has more than one direction? 50 Km in this direction 70 Km in this direction This is called the RESULTANT vector A B C Journey C is the same as taking A then B – the vector equation is A + B = C
5. 5. Vectors and France <ul><li>Use the blank map of France and the distance table to draw the main towns on the French map. </li></ul>
6. 7. Pythagoras’ Theorem <ul><li>We can use Pythagoras Theorem to calculate resultant vectors </li></ul>Do the distances on your map agree with this?
7. 8. Proving Pythagoras Theorem <ul><li>Cut a piece of square paper into 3 squares: </li></ul><ul><ul><li>One 3 x 3 squares </li></ul></ul><ul><ul><li>One 4 x 4 squares </li></ul></ul><ul><ul><li>One 5 x 5 Squares </li></ul></ul><ul><li>Fit them together to make a right-angled triangle. </li></ul><ul><li>Count the number of squares on each side of the equation: </li></ul><ul><li>a 2 + b 2 = c 2 </li></ul>
8. 9. Speed and Velocity Speed = Distance Travelled / Time Taken = m/s IT IS A SCALAR QUANTITY – IT ONLY HAS MAGNITUDE Velocity = Distance Travelled / Time Taken = m/s IT IS A VECTOR – IT HAS MAGNITUDE AND DIRECTION Distance Speed x Time
9. 10. Using the speed triangle, try these… <ul><li>1. You are watching a batsman hit a cricket ball. If 0.375 seconds passes between the time you see him strike the ball and the time you hear the sound of this, how far from the batsman are you sitting? The speed of sound in air is 340 m s–1. (The speed of light is nearly a million times bigger than this, so you see the bat hit the ball more or less at the instant it occurs.) </li></ul><ul><li>2. A girl diving from a 15 m platform wishes to know how fast she enters the water. She is in the air for 1.75 s and dives from rest (with an initial speed of zero). What can you tell her about her entry speed? </li></ul><ul><li>3. An experiment performed on the Moon finds that a feather falls 20.75 m from rest in 5 s. What is its speed as it hits the Moon's surface? </li></ul>
10. 11. Using the speed triangle, try these… <ul><li>1. Distance = speed x Time = 340 x 0.375 = 127.5m </li></ul><ul><li>2. Average Speed = Distance / Time = 15 / 1.75 = 8.57m/s </li></ul><ul><li>As started at zero, must be at twice average when enters water = 17.1m/s </li></ul><ul><li>3. Average Speed = Distance / Time = 20.75 / 5 = 4.15m/s </li></ul><ul><li>Hence maximum Speed = 8.30m/s </li></ul>
11. 12. Distance Time Graph <ul><li>If a graph shows distance against time, what can we tell from it..? </li></ul>Time (s) Distance (m)
12. 13. Distance Time Graph <ul><li>If a graph shows distance against time, what can we tell from it..? </li></ul>Time (s) Distance (m) A B C D Describe each section, A to D
13. 14. Distance Time Graph Time (s) Distance (m) Moving at constant speed Stationary Constant speed but faster than A Moving back towards start point What else can we calculate from the graph?
14. 15. Plot a Distance Time Graph for 1 marble journey…
15. 16. Distance Time Graph Time (s) Distance (m) Gradient = Distance /Time = Speed Or dy / dx But it is different at different places?
16. 17. Instantaneous Speed Time (s) Distance (m) What is the speed here? Calculate the gradient of the TANGENT on your graph The Gradient at this point is the INSTANTANEOUS SPEED
17. 18. Average Speed Time (s) Distance (m) What is the AVERAGE Speed for the whole journey? Calculate the gradient of the CHORD on your graph The Gradient of the CHORD is the AVERAGE SPEED
18. 19. Try this What is the instantaneous speed here
19. 20. Try this
20. 21. Try this What is the average speed between 40 and 60 mins?
21. 22. Try this
22. 23. Questions to Try <ul><li>Page 181 – show working to get the answer at the bottom: </li></ul><ul><li>1, 2, 3, 4, </li></ul><ul><li>5 (note there are 1.6 km per mile) </li></ul><ul><li>6,7 </li></ul>
23. 24. Velocity Time Graphs <ul><li>Different to Distance Time – y axis is Velocity not Distance!!!! </li></ul><ul><li>Be careful not to get them mixed up! </li></ul>
24. 25. Velocity Time Graph <ul><li>If a graph shows velocity against time, what can we tell from it..? </li></ul>Time (s) Velocity (m/s) Speeding up – constant acceleration Constant Velocity Constant Acceleration Slowing Down - decelerating Stationary
25. 26. Write these into your book with answers… For each of the graphs in question 2, calculate the distance travelled overall.
26. 27. To recap on last lesson, answer these in the back of your book… Distance travelled is area under graph…
27. 28. Making a Velocity Time Graph…
28. 29. Velocity Time Graph <ul><li>The area under the graph is the total distance travelled </li></ul>Time (s) Velocity (m/s)