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Gradient

Uniform motion. The following symbols will be used throughout M1:. Displacement (distance). Initial velocity. Consider a velocity-time graph of an object moving with these variables:. Final velocity. Acceleration. Now consider the gradient and area under the line. Time.

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Gradient

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  1. Uniform motion The following symbols will be used throughout M1: Displacement (distance) Initial velocity Consider a velocity-time graph of an object moving with these variables: Final velocity Acceleration Now consider the gradient and area under the line Time Eg a car accelerates at a constant rate from 10ms-1 to 26ms-1 in 8 seconds. Calculate the acceleration and distance travelled during this period. = 2 ms-2 Gradient Area = 144 m

  2. Eg A racing car is travelling on a straight horizontal road. Its initial speed is 25 m s–1 and it accelerates for 4 s to reach a speed of V m s–1. It then travels at a constant speed of V m s–1 for a further 8 s. The total distance travelled by the car during this 12 s period is 600 m. (a) Sketch a speed-time graph to illustrate the motion of the car during this 12s period. (b) Find the value of V. Area = distance B Area A But area = 600 m (c) Find the acceleration of the car during the initial 4 s period.

  3. WB1 A sprinter runs a race of 200 m. Her total time for running the race is 25 s. The diagram above is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of 9 m s–1 in 4 s. The speed of 9 m s–1 is maintained for 16 s and she then decelerates uniformly to a speed of u m s–1 at the end of the race. Calculate (a) the distance covered by the sprinter in the first 20 s of the race, 2. Area Area = distance (b) the value of u, Remaining distance = 38 (c) the deceleration of the sprinter in the last 5 s of the race.

  4. WB2 A train starts from rest at a station A and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s–2. For the next 24 s it moves with constant acceleration 0.75 m s–2. It then moves with constant speed for T seconds. Finally it slows down with constant deceleration 3 ms–2 until it comes to rest at a station B. (a) Show that, 34 s after leaving A, the speed of the train is 30 m s–1. (c) Find the distance moved by the train during the first 34 s of its journey from A. After 10s, After 34s, (b)Sketch a speed-time graph to illustrate the motion of the train as it moves from A to B. Area The distance from A to B is 3 km. (d) Find the value of T. Total distance = 3000 m Distance travelled while decelerating

  5. Acceleration-time graphs Eg A car is travelling on a straight horizontal road. Below is a speed-time graph for its journey. Sketch an acceleration-time graph to illustrate the motion of the car Work out the acceleration for each stage of the journey

  6. Distance-time graphs Consider these speed-time graphs: Area = distance Accelerating Constant speed Deccelerating Distance = 60m Distance = 72m Distance = 75m Now consider what distance-time graphs for these would like: An accelerating object will gain distance at an increasing rate An object moving with constant speed will gain distance at a constant rate A decelerating object will gain distance at a decreasing rate

  7. Distance-time graphs Eg A car is travelling on a straight horizontal road. Below is a speed-time graph for its journey. Sketch a distance-time graph to illustrate the motion of the car Work out the distance for each stage of the journey

  8. Equations of uniform motion You have already met the following rules of a particle moving with constant acceleration: (2) (1) Using these, further rules which are more useful in some problems, can be found: Make t the subject of (1): Make v the subject of (1): (3) Substitute this in (2): Substitute (3) in (2): (5) (4)

  9. Equations of uniform motion These last 3 rules are the ones you will most commonly use Eg a car is travelling along a straight road with a constant acceleration of 0.75ms-2. The car is travelling at 8ms-1 when it passes a man, 12 seconds later it passes a dog. Find (a) the distance between the man and the dog (b) the speed with which the car passes the dog Information given: Decide which rule to use based on the information given Eg a sprinter accelerates at a constant rate from 5 m s-1 to 17 m s-1. They cover 44 m in this time. Find their acceleration You may need to make the unknown you require the subject of a rule Information given:

  10. WB3 An aircraft moves along a straight horizontal runway with constant acceleration. It passes a point A on the runway with speed 16 m s–1. It then passes the point B on the runway with speed 34 m s–1. The distance from A to B is 150 m. (a) Find the acceleration of the aircraft. Information given: (b) Find the time taken by the aircraft in moving from A to B. Information given: You could use But this leads to a quadratic equation which is harder to solve (c) Find, to 3 significant figures, the speed of the aircraft when it passes the point mid-way between A and B. Information given:

  11. WB4 Two cars A and B are moving in the same direction along a straight horizontal road. At time t = 0, they are side by side, passing a point O on the road. Car A travels at a constant speed of 30 m s–1. Car B passes O with a speed of 20 m s–1, and has constant acceleration of 4 m s–2. Find (a) the speed of B when it has travelled 78 m from O, Information given: (b) the distance from O of A when B is 78 m from O, Car B at this time: Car A at this time: (c) the time when B overtakes A.

  12. Deceleration If a particle is slowing down it has negative acceleration, called deceleration. Eg a car slows down at a constant rate from 40 m s-1 to 12 m s-1 in 4 seconds. Find the deceleration and distance travelled in this time. Eg a car is travelling at 20 m s-1 when a cat runs out into the road 80m away. Calculate the minimum deceleration needed to avoid hitting the cat.

  13. Information given: 13. Ex 5B a) b) Information given: 14. a) b ) furthest point from A must be when This is 2m further than A, so distance travelled must be 2 x 2 = 4m

  14. a) For BC: 15. 300m 100m B A C b) For AB: Now for AC: 16a) and b) c)

  15. Acceleration due to gravity Ignoring air resistance, any object, regardless of mass, accelerates towards the Earth. The rate of acceleration is denoted by the letter g and is approximately 9.8 m s-2 Eg A skydiver drops from a plane and accelerates under gravity for 10 seconds. Find (a) the distance travelled in this time (b) his velocity after 10 seconds Information given: It is sometimes useful to leave expressions in terms of g Eg Another skydiver drops from a plane and reaches 44.1 m s-1 Find (a) the distance travelled in this time (b) the time he has been falling Information given:

  16. Eg a ball B is projected vertically upwards with speed 15 m s-1 from a point P. Find (a) the greatest height above P reached by B (b) the total time before B returns to P Greatest height There is a second solution t = 0, but t > 0 for any real scenario Positive direction

  17. WB5 A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of 4 m s-1 upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find (a) her speed when she reaches the surface of the pool, (c) State two physical factors which have been ignored in the model. Air resistance The size (length) of the diver Positive direction (b) the time taken to reach the surface of the pool.

  18. WB6 A stone is thrown vertically upwards with speed 16 m s–1 from a point h metres above the ground. The stone hits the ground 4 s later. Find (a) the value of h, (b) the speed of the stone as it hits the ground

  19. Ex 2E Q16a) Information given: b) Information given: More information needed – get acceleration using (a)

  20. Q17 Between B and M: M B Between A and B: A

  21. Assignment feedback Q2. A small ball is projected vertically upwards from ground level with speed u. The ball takes 4s to return to ground level. Draw a velocity-time graph to represent the motion of the ball during the first 4s Q3a) A firework starts from rest & moves vertically, covering 27m in 3s. Find a and v after 3s a is constant Velocity Q3. After 3 seconds, the rocket burns out & the firework moves freely under gravity. Find the height reached after 5 seconds. For , Total distance = 27 + 16.4 = 43.4m

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