# Equations of motion

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Education

Published on March 30, 2013

Author: LKOTZE

Source: slideshare.net

## Description

Based on Andries Oliviers' CAPS aligned textbook

Equations of motion

Motion in one dimension

What is motion?Motion is when an object changes position.How do you know that the race car moved? ◦ It changed its position on the track.

How can you tell something haschanged position? Inorder to see if something has changed position (motion) you need a reference point. ◦ For example, the starting or finishing line of a racetrack. Click to edit Master text styles Second level Third level Fourth level Fifth level

PositionIfsomething moves, it constantly changes position.When you describe position, you refer to a point of reference/origin (zero).When you have chosen your point of reference all positions will be chosen relative to this point of origin.Position is the place where an object is as observed from a point of reference.

Frame of reference Point of reference -3 m/3m to the left/ 3m W +5 m /5m to the right/ 5m E X-axisLinear movement:when we need only one axis. Y-axis (not used simultaneously with x-axis in grade 10)

GPS24 satellitesAt least 4 sattelites will be visible at any time from any pointEach sattelite sends a signal to GPS receiversThe position of the receier can be determinedPosition is given in latitude, longitude and height

Distance anddisplacement

What is distance?Easy question!Distance is how far an object has moved.We measure distance in METERS!Distance is the actual path length that is taken.

Distance vs Displacement B Distance Displacement A

What is displacement?Suppose a runner jogs north to the 50-m mark and then turns around and runs back south 30-m.Total distance is 80-m.Two directions - north and south.Displacement is the distance and direction.

DisplacementDisplacement is the change in position of the object.In other words, a straight line.The magnitude of the displacement will be smaller than or equal to the distance that was covered.Displacement will be represented by Δx on the horisontal line, and Δy on the vertical.

Calculating distance and displacement Δx = xf – xi = _m Let’s try this: You move from your front porch to your neighbours’ house 400 m away. Now calculate your displacement. Moving from your front door (xi = 0 m ), and to your neighbours’ house (xf = 400 m). Δx = xf – xi = 400 – 0 = 400 m to the right

Calculating distance and displacement II Now, you move from your neighbours’ house and move back towards your house, but move beyond and travel to the cafe 600 m from your home. Now calculate your displacement from the moment you left home. Moving from your front door (xi = 0 m ), and to the cafe (xf = 600 m). Δx = xf – xi = 600 – 0 = 600 m to the left.

Scalars and vectorsScalars are quantities that has only magnitude.Examples: distance time mass volume, energy, work and potential differenceVectors are quantities that posesses both magnitude and direction.Examples: displacement velocity acceleration force weight

Speed and velocity

What is speed?Speed is the distance an object travels over time.Any change over time is called a RATE.Speed is the rate at which distance is traveled.

SPEED FORMULA Speed = Distance / Time v = Δx / t Click to edit Master text styles Second level Third level Fourth level Fifth level

Speed Example  Suppose you ran 2 km in 10 min. ◦ What is your rate? v = Δx / t v= 2 km /10 min. v= 0.2 km/min. Click to edit Master text styles Remember the units! Second level Third level Fourth level Fifth level

Constant Speed…What does constant mean?Ifyou are driving on the highway and you set your cruise control, you are driving at a constant speed.What would a constant speed graph look like?

Constant Speed Graph

Do you always go the same speed? No! Most of the time you are increasing speed, decreasing speed, or stopping completely! Think about driving a car or riding a bike! Click to edit Master text styles Second level Third level Fourth level Fifth level

What would a changing speed graph looklike?

What is average speed?How do you find an average?Average speed is the total distance traveled over the total timev = Δxtotal / ttotal

What is Instantaneous speed?What does a speedometer in a car do? ◦ It shows how fast a car is going at one point in time or at one instant.Instantaneous speed is the speed at a given point in time.

What is Velocity?Speed is how fast something is moving.Velocity is how fast something is moving and in what direction it is moving.Why is this important? ◦ Hurricanes ◦ Airplanes

Speed or Velocity? Ifa car is going around a racetrack, its speed may be constant (the same), but its velocity is changing because it is changing direction. Click to edit Master text styles Second level Third level Fourth level Fifth level

Speed or Velocity? Escalators have the same speed (constant), but have different velocities because they are going in different directions. Click to edit Master text styles Second level Third level Fourth level Fifth level

acceleration

AccelerationWhen an objects velocity changes, it accelerates.Acceleration shows the change in velocity during a period of time.Acceleration = change in velocity / time a = Δv/Δt = vf - vi tf-tim·s-2

Acceleration IIMagnitude is calculated using the formula.The direction can be determined as long as motion is in one dimension.v = increase, a in the same direction.v = decrease, a in opposite direction.

Positive accelerationPositive values show that the ais in the same direction as themotion. An increase in v. Negative acceleration Negative values show that the a is in the opposite direction from the motion. A decrease in v. Constant or uniform acceleration We only use constant acceleration. Because we work with constant a, instantaneous and average acceleration will always be the same. This means we use the same formula to calculate all three types of acceleraion.

Describing motion withdiagrams

What is a motion diagram? Click to edit Master text styles ◦ Second level  Third level  Fourth level  Fifth level

Images are equal distances apart. Object occupies a single position. Object is at rest. Constant velocity.Increase in distance between images. Decrease in distance between images. Moving faster, accelerating. Moving slower, decelerating.

Sketches and usesSketch a row of dots to represent the object.Refer to previous slide to get the spaces right.Draw a vector from each dot to show velocity.If acceleration occurs, add one more vector to illustrate acceleration.

The first dot is always labeled zero, and the time elapsedBetween the dots are the same throughout. A B C D E 0 1 2 3 4 ΔtAB = ΔtBC = ΔtCD = ΔtDE

Ticker timer and ticker tape Frequency: number of dots that the timer’s hammer makes each second. Frequency of 50 Hz. Period: the time it takes from dot to the next. 1/50 = 0,02s. Time interval: describes a collection of periods. Done in 1-dot, 5-dot or 10-dot intervals.

Determine the magnitude of the avgvelocityYou will need Δx and Δt for each specific interval.Δx = determined by measuring spaces between intervals.Δt = determined by multiplying the spaces with the period.

ExampleIn the first tape, John is moving steadily while pulling aticker tape. Calculate John’s average velocity.

Determining the avg accelerationYou need four sets of information:vf, vi, tf and tivf – vi = Δv for first interval, and for last interval.tf – ti = Δt where time is relative to the time in the centre of the first and the last interval.

Example Click to edit Master text styles ◦ Second level  Third level  Fourth level  Fifth levelIn the third tape, Sarah walks faster, while pullinga ticker tape. Calculate Sarah’s average acceleration.

Description of motionusing graphs

Calculus – the abridged edition Slope of the line (derivative) Displacement Velocity acceleration Area under the curve (integral)

Movement and velocity

Positive velocity, positive direction from rest.v(m·s-1) t (s)

v(m·s-1) t (s)

v(m·s-1) t (s)

v(m·s-1) t (s)

v(m·s-1) t (s)

v(m·s-1) t (s)

Position-time graphs for velocity

v(m·s-1) t (s)Δx (m) t (s)

Movement with acceleration

v(m·s-1) t (s)Δx (m) t (s)a(m·s-2) t (s)

Finding velocity ∆x + 8 m Slope = = = +4 m s ∆t 2s

Instantaneous Velocity

Finding acceleration ∆v + 12 m s Slope = = = +6 m s 2 ∆t 2s

Finding displacement v v – vo = at area= lxb + ½ bxh vo Δx= vit + ½ t(at) velocity Δx = vt + ½ at2 t time

Equations of motion

Symbolsvi = initial velocity in m·s-1vf = final velocity in m·s-1a = acceleration in m·s-2Δx or Δƴ = displacement in mt = time in s

The formulas vf = vi +aΔt Δx = viΔt + ½ aΔt2 Δx = (vf2+ vi)Δt Vf2 = vi2 + 2aΔx

PrinciplesWhen using v, a and Δx, remember that they are vectors, so take direction into account. Original direction of motion = +.If an object starts from rest, vi = 0 m/sIf an object stops, vf = 0 m/sIf chosen direction is +, then all v, and Δx substitutions are +. If velocity increases, it is +.If velocity decreases, it is -.Equations can only be used for motion with constant acceleration in a straight line.Work in SI units.

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