1 Motion in 1D o Frames of Reference o Speed  average  instantaneous o Acceleration o Speed-time graphs and distance travelled Physics -I Piri Reis University

2 Speed o Distance is the number of metres between two points o Time is the number of seconds it takes for something to happen o If a person walks a distance x metres in t seconds, then we define the persons walking speed, v, to be x /t m/s v = x /t o Strictly this is their AVERAGE speed whilst walking. o The distance travelled is then d = vt

3 Speed o If only a short part of the walk is considered, say x 2 and that part takes a time t 2 to walk, then o the speed for that part of the walk could be different o Slower o or faster o If we keep reducing the size of x 2, then o the time t 2 will also get shorter o If we keep reducing x 2 until it becomes ~0, then o the time t 2 will also be ~0 o The ratio x 2 /t 2 is the INSTANTANEOUS speed. o This is then the derivative of the distance wrt t v = d x /dt ≈ x 2 /t 2 x x2x2

4 Speed o Instantaneous speed is what is read from a car speedometer o Average speed is what matters for a long trip. o The average speed for each part of the trip is d i /t i o The average speed for the trip v = { ∑ x i } / { ∑t i } o The average speed is NOT s = { ∑ x i /t i } / 6 o because the distances are not necessarily equal d x6x6 x5x5 x4x4 x3x3 x2x2 x1x1

5 Time, t Distance, x On a graph of time vs distance instantaneous speed is the slope The person started to walk at t = 0 from x = 0 Each point on graph shows Where they are at a given time Distance-time graphs

6 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Person walks with constant speed Distance, x

7 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Person walks with slowing speed Distance, x

8 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Person walks with increasing speed Distance, x

9 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Person speeds up and then slows down Distance, x

10 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Person walks with constant speed then stops Distance, x

11 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Person stands still walks at constant speed then stops and stands still Distance, x

12 time On a graph of time vs distance instantaneous speed is the slope Distance-time graphs Here the person changed Direction and walked backwards And forwards again Distance, x

13 Time, t Distance, x Distance-time graphs At any point on the curve, the tangent is The instantaneous speed, v = d x /dt When the slope is negative the direction is backwards

14 Acceleration o To change speed, the walker must accelerate o The average acceleration, a, is (Total increase in speed) (Time taken to change) o The instantaneous acceleration is the limit as the changes become small, just like speed instantaneous a = dv/dt = d 2 x /dt 2 a =

15 Time, t Distance, x Acceleration on a distance-time graph At any point on the curve, the tangent is The instantaneous speed, v = d x /dt The curvature of the line tells about acceleration This curvature is deceleration This is acceleration

16 Time, t Distance, x Acceleration on a distance-time graph At any point on the curve, the tangent is The instantaneous speed, v = d x /dt The curvature of the line tells about acceleration Zero acceleration deceleration acceleration Slope increasing Slope decreasing

17 Time, t Speed, d x /dt Speed-time graph At any point on the curve, the tangent is The instantaneous acceleration, a = d 2 x /dt 2 This person is walking at constant speed (Acceleration is zero)

18 Time, t Speed, v = d x /dt Speed-time graph At any point on the curve, the tangent is The instantaneous acceleration, a = d 2 x /dt 2 This person is accelerating Constantly

19 Time, t Speed, d x /dt Speed-time graph At any point on the curve, the tangent is The instantaneous acceleration, a = d 2 x /dt 2 This person is accelerating but the rate of acceleration is decreasing

20 The area under the curve in a speed-time plot is the distance travelled Time, t Speed, d x /dt Speed-time graph Recall x = vt for constant speed which, more generally, is x = ∫ d x /dt dt

21 Time, t Speed, d x /dt Speed-time graph x = ∫ d x /dt dt x = 0.5 x t 0 x s max t0t0 s max In this case

22 Time, t Speed, d x /dt Speed-time graph d = ∫ d x /dt dt In this case the curve can be broken into several triangles & squares to work out the distance travelled

23 Time, t Speed, d x /dt Speed-time graph x = ∫ d x /dt dt t0t0 In this case the curve can be broken into several triangles & squares to work out the distance travelled t1t1 t2t2 t3t3 t4t4 t5t5 s4s4 s3s3 s2s2 s1s1

24 Time, t Speed, dx/dt Speed-time graph x = ∫ d x /dt dt t0t0 In this case - good luck!

25 Reference Frames o What something looks like depends on where you look from  big when close  small when far away o In physics it is very important to be clear about where you are looking from,  where you are sitting when you make a measurement,  Where you are imagining you are when you do a calculation o But what is important is not the distance, but the speed of where you are looking from.  We call the person who is looking ‘the observer’,  We call the place they are looking from ‘the reference frame’  It is called a ‘frame’ because we imagine a coordinate system with three axis, which looks like a ‘frame’.

26 Frames can have Different origins If the frames are arranged to Be in the same orientation Then x 1 = x 2 + offset distance

27 Frames can have Different rotations

28 Frames can have Different speeds

29 Different speeds is the most significant for what we will be doing

30 Reference Frames o A reference frame is a ’place to look from’ o It is the coordinate system we are measuring from The speed of the blue car depends on where it is measured from Measured from the road it is V 1 Measured from the green car it is V 1 -V 2

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32 Z.Akdeniz Reference Frames o In an elevator accelerating up the effective gravity is increased o Accelerating down the effective gravity is reduced o At a constant speed up or down there is no difference  This is a reference frame which moves with constant speed o If the elevator is in free fall, then there is ‘no gravity’ and the frame of reference is effectively an inertial frame  In fact it is accelerating, but then every frame is - gravity is everywhere in the universe

33 Reference Frames o More about gravity in a couple of weeks.