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Motion in 1D

o Frames of Referenceo Speed

average instantaneous

o Accelerationo Speed-time graphs and distance travelled

Physics -IPiri Reis University 2010-2011

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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

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Speed

o If only a short part of the walk is considered, say x2 and that part takes a time t2 to walk, then

o the speed for that part of the walk could be differento Slower o or faster

o If we keep reducing the size of x2, theno the time t2 will also get shorter

o If we keep reducing x2 until it becomes ~0, theno the time t2 will also be ~0

o The ratio x2/t2 is the INSTANTANEOUS speed.o This is then the derivative of the distance wrt t

v = dx/dt ≈ x2/t2

x

x2

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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 di/ti

o The average speed for the trip

v = { ∑xi } / { ∑ti }

o The average speed is NOT s = { ∑xi/ti } / 6o because the distances are not necessarily equal

d

x6x5x4x3x2x1

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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 showsWhere they are at a given time

Distance-time graphs

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Person walks withconstant speed

Distance, x

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Person walks withslowing speed

Distance, x

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Person walks withincreasing speed

Distance, x

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Person speeds upand then slows down

Distance, x

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Person walks withconstant speed then stops

Distance, x

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Person stands stillwalks at constant speedthen stops and stands still

Distance, x

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time

On a graph of time vs distance instantaneous speed is the slope

Distance-time graphs

Here the person changedDirection and walkedbackwards

And forwards again

Distance, x

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Time, t

Distance, x

Distance-time graphs

At any point on the curve, the tangent isThe instantaneous speed, v = dx/dt

When the slope is negative the direction is backwards

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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 = d2x/dt2

a =

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Time, t

Distance, x

Acceleration on a distance-time graph

At any point on the curve, the tangent isThe instantaneous speed, v = dx/dtThe curvature of the line tells about acceleration

This curvature is deceleration

This is acceleration

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Time, t

Distance, x

Acceleration on a distance-time graph

At any point on the curve, the tangent isThe instantaneous speed, v = dx/dtThe curvature of the line tells about acceleration

Zeroacceleration

deceleration

accelerationSlope increasing

Slope decreasing

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Time, t

Speed, dx/dt

Speed-time graph

At any point on the curve, the tangent isThe instantaneous acceleration, a = d2x/dt2

This person is walking atconstant speed (Acceleration is zero)

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Time, t

Speed, v = dx/dt

Speed-time graph

At any point on the curve, the tangent isThe instantaneous acceleration, a = d2x/dt2

This person is acceleratingConstantly

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Time, t

Speed, dx/dt

Speed-time graph

At any point on the curve, the tangent isThe instantaneous acceleration, a = d2x/dt2

This person is acceleratingbut the rate of accelerationis decreasing

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The area under the curve in a speed-time plot is the distance travelled

Time, t

Speed, dx/dt

Speed-time graph

Recall x = vt for constant speedwhich, more generally, is

x = ∫ dx/dt dt

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Time, t

Speed, dx/dt

Speed-time graph

x = ∫ dx/dt dt

x = 0.5 x t0 x smax

t0

smax In this case

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Time, t

Speed, dx/dt

Speed-time graph

d = ∫ dx/dt dt

In this case the curvecan be broken into severaltriangles & squares to work out the distance travelled

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Time, t

Speed, dx/dt

Speed-time graph

x = ∫ dx/dt dt

t0

In this case the curvecan be broken into severaltriangles & squares to work out the distance travelled

t1 t2 t3 t4 t5

s4

s3

s2

s1

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Time, t

Speed, dx/dt

Speed-time graph

x = ∫ dx/dt dt

t0

In this case - good luck!

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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’.

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Frames can haveDifferent origins

If the frames are arranged toBe in the same orientationThen x1 = x2 + offset distance

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Frames can haveDifferent rotations

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Frames can haveDifferent speeds

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

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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 fromMeasured from the road it is V1

Measured from the green car it is V1-V2

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Reference Frames

o A speed or distance must have a frame of reference

o Frames of reference may be ‘inertial’ which are frames where objects without any forces on them stay in constant motion. All inertial frames move at constant speed with respect to each otherAll inertial frames move at constant speed with respect to each othero Or a frame may be non-inertial, where if an object has no force on it, it will still accelerate.

o Inertial frames are very rare on earth [free fall elevator]o but we usually consider the earth to be inertial and everything has a force on it - gravity

32Z.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

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Reference Frames

o More about gravity in a couple of weeks.