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# 2.1 linear motion

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An overview to motion in a straight line (kinematics)

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### 2.1 linear motion

1. 1. Topic 2 - Mechanics 2.1 - Kinematics
2. 2. Velocity, Speed, Acceleration <ul><li>Kinematics is the study of linear motion.
3. 3. There are a number of key terms to define; </li><ul><li>Displacement – This is the distance from a given starting point to the current position in a straight line.
4. 4. Velocity – This is the speed of an object in a given direction </li><ul><li>Velocity is the rate of change of position </li></ul><li>Acceleration – This is the rate of change of velocity in a given direction.
5. 5. All three of these quantities are vectors. </li></ul></ul>
6. 6. Average Speed and Velocity <ul><li>The average speed of an object is defined as: </li></ul><ul><ul><li>NB. It is the total distance and the total time that are important. </li></ul><li>The average velocity of an object is defined as: </li></ul>
7. 7. Average Speed and Velocity <ul><li>Calculate the average speed and velocity of: </li><ul><li>A car that travels at 30ms -1 East for 2 minutes before turning North and travelling at 40ms -1 for 5 minutes.
8. 8. A ball that travels at 10ms -1 for 15s before striking a wall and rebounding at 45 o to the original path at 8ms -1 for 5s.
9. 9. A 400m runner on a standard athletics track that completes the race in 52s. </li></ul></ul>
10. 10. Instantaneous Motion <ul><li>Average values of displacement, velocity and acceleration give reasonable approximations of the behaviour of a system. </li><ul><li>They tell you what the system was like before and what it will be like afterwards. </li></ul><li>However, they do lack detail. </li><ul><li>They do not tell you how and object got from A to B only that it did. </li></ul><li>Instantaneous values for displacement, velocity and acceleration give a much clearer picture. </li></ul>
11. 11. Instantaneous Motion <ul><li>Instantaneous displacement (position) is often simply measured. </li><ul><li>It can be recorded by various means including using cameras. </li></ul><li>Instantaneous velocity is calculated from a displacement time graph. </li><ul><li>As velocity is the rate of change of position it is the gradient of a displacement time graph </li></ul><li>Instantaneous acceleration is calculated from a velocity time graph. </li><ul><li>As acceleration is defined as the rate of change of velocity, it is found as the gradient of a velocity time graph. </li></ul></ul>
12. 12. Motion Graphs <ul><li>Plot the following displacement-time data on a graph.
13. 13. Take suitable measurements to construct the corresponding velocity-time and acceleration-time graphs </li></ul>s /m 20.00 33.75 45.00 53.75 60.00 63.75 65.00 63.75 60.00 53.75 45.00 33.75 20.00 3.75 t /s 0 1 2 3 4 5 6 7 8 9 10 11 12 13
14. 14. Uniform Acceleration <ul><li>In the special case where the acceleration on an object is constant a number of equations can be derived to describe the object's motion.
15. 15. Consider the velocity-time graph for this object.
16. 16. Let the object change velocity from (t 1 ,u) to (t 2 ,v) </li></ul>v t (t 1 ,u) (t 2 ,v)
17. 17. Uniform Acceleration <ul><li>By definition: </li><ul><li>The acceleration is the rate of change of velocity. </li></ul></ul>v t (t 1 ,u) (t 2 ,v)
18. 18. Uniform Acceleration <ul><li>By definition: </li><ul><li>The average velocity is the total displacement over time taken </li></ul></ul>v t (t 1 ,u) (t 2 ,v)
19. 19. Uniform Acceleration <ul><li>By definition: </li><ul><li>The total displacement is the area under the curve. </li></ul></ul>v t (t 1 ,u) (t 2 ,v)
20. 20. Uniform Acceleration <ul><li>By definition: </li><ul><li>The total displacement is the area under the curve. </li></ul></ul>v t (t 1 ,u) (t 2 ,v)
21. 21. Uniform Acceleration <ul><li>By eliminating t from equations 1 and 2: </li></ul>v t (t 1 ,u) (t 2 ,v)
22. 22. Uniform Acceleration - Summary <ul><li>The 5 equations that can be used when a is constant are: </li></ul>v t (t 1 ,u) (t 2 ,v)
23. 23. Practice <ul><li>An car has initial velocity 20ms -1 . It accelerates for 4.67s at a rate of 2.79ms -2 . What is its new velocity and what distance was covered whilst it accelerated?
24. 24. A ball is rolled across a rough table and covers a distance of 2.37m in 16.3s before coming to rest. What is the acceleration on the ball due to the friction (assume constant) and what was the ball's initial speed? </li></ul>
25. 25. Practice <ul><li>A cannon is fired horizontally. The shell has a constant deceleration of 0.05ms -2 and travels a distance of 1370m before hitting its target with speed 243ms -1 . How long is the cannon ball in the air and with what speed did it leave the barrel of the gun. (ignore gravity)
26. 26. A ball is kicked across smooth ground at a wall 15m away with speed 20ms -1 . The ball rebounds with speed 16ms -1 . The ball squashes by 20mm when in contact with the wall. Calculate the average acceleration exerted by the wall on the ball. </li></ul>
27. 27. <ul><li>An object falling freely (no forces acting except its weight) in a vacuum close to the Earth's surface will experience a downward acceleration of g =9.81ms -2 .
28. 28. This is the same value as the Earth's gravitational field strength and is due to the gradient of the gravitational field at this point.
29. 29. In Physics we approximate any small, smooth object falling in air as having an acceleration of 9.81ms -2 </li></ul>Free Fall
30. 30. Practice <ul><li>A ball is dropped from rest at a height of 3m. Calculate the velocity of the ball just before it strikes the ground and the time taken to fall.
31. 31. A ball is thrown vertically upwards with an initial speed of 2.5ms -1 . Calculate the maximum height of the ball and the total time of flight.
32. 32. A 2kg ball rolls down a smooth slope that is 5m long. Calculate the speed of the ball as it reaches the bottom of the slope. </li></ul>
33. 33. Terminal Velocity <ul><li>For real objects falling in viscous fluids the object suffers a resistive force that is (usually) proportional to the objects speed relative to the fluid.
34. 34. This means that as the object travels faster, the resistive forces also increase.
35. 35. At some point the resistive forces will be equal in magnitude to the weight force acting on the object. </li><ul><li>There will be no nett downwards force.
36. 36. There will be no downwards acceleration.
37. 37. There will be no increase in velocity. </li></ul><li>This maximum speed is called an object's terminal speed. </li></ul>
38. 38. Relative Velocity <ul><li>Imagine two objects (A and B) that are both moving.
39. 39. What does the velocity of A look like compared to the Universe?
40. 40. What does the same velocity look like when observed from B?
41. 41. The motion of two objects relative to each other is often of great interest to Physicists as this is how we observe the Universe in reality. </li><ul><li>We are never entirely stationary so we need to be able to calculate relative velocities. </li></ul></ul>
42. 42. Relative Velocity <ul><li>The velocity of A as observed from B (the velocity of A relative to B (A/B)) is defined as: </li></ul><ul><li>Here O represents some fixed co-ordinate system that other measurements can be made against.
43. 43. Be careful with calculations: v is a vector so direction must be accounted for! </li></ul>
44. 44. Practice <ul><ul><ul><li>Car A is travelling with velocity 70ms -1 due north.
45. 45. Car B is travelling with velocity 35ms -1 due north.
46. 46. Car C is travelling with velocity 50ms -1 due South.
47. 47. Car D is travelling with velocity 45ms -1 30 o South of East. </li></ul></ul><li>Calculate: </li><ul><li>The velocity of A relative to B
48. 48. The velocity of A relative to C
49. 49. The velocity of B relative to C
50. 50. The velocity of C relative to B
51. 51. The velocity of D relative to C
52. 52. The velocity of A relative to D </li></ul></ul>