84A - Describing Motion

Mechanics is the branch of physics concerned with the behaviour of objects when subjected to forces and the subsequent effects of these objects on other objects. Scientists such as Galileo, Kepler, and especially Newton, laid the foundation for what is now known as classical mechanics. It is a branch of classical physics that deals with the particles that are moving either with less velocity or that are at rest.

This section of work covers kinematics, the branch of classical mechanics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion (which are gerenally forces). To describe motion, kinematics studies the paths of objects and their differential properties such as velocity and acceleration.

In this section of work the motion of objects moving in one dimension is described using the quantities distance, displacement, speed, velocity and acceleration. Students are introduced to some equations used to predict aspects of motion at specified times in an object's path and also to the representation of such motion in graphs.

Objects travelling in a circle at a constant speed are used to demonstrate motion in two dimensions and the concept of centripetal acceleration is also introduced.

84A1 - Distance, Time and Speed

Here you will learn to:

Identify the scalar quantities used to describe and measure motion by their symbols and units,

Describe speed as the rate of change of distance,

Use an equation to calculate either speed, distance or time for objects moving in a straight line, and

Distinguish between average and instantaneous speed.

Scalars

When we measure things in Physics we use quantities such as length, mass and time. All measurement quantities can be classified into two groups: scalars that have magnitude (size) only and vectors that have both magnitude and direction. Time is an example of a scalar quantity because it can only go in one direction, which is forwards. If we say that it took three hours to drive from Sydney to Canberra, we do not need to indicate that it was three hours into the future because time is a scalar and, therefore, we all understand that it goes forwards only. Mass, temperature and energy are other examples of scalar quantities as they have no direction associated with their measurements.

Speed

Everyone seems to have a good intuitive understanding of what speed is by describing how fast things are going. Physics is a mathematical and science and uses the rate concept to precisely define the speed of any moving object. The rate of change of any quantity is simple how much of it is happening in a given time frame. For example, a tap might eject 150 mL of water each second so it's flow rate would be 150 milliliters per second (mL s-1). For the average 10-year-old, the growth rate is about 4 millimeters per month (mm month-1). This means that in their tenth year their height would increase by about 4 mm each month or 48 mm in a whole year. The growth rate can increase or decrease depending on how old the person is. In the first year of life, a baby can grow at an astonished fast rate of 36 mm month-1 and at the age of 16 it drops to a rate of only 0.5 mm month-1 and to roughly zero at age 20.

It should be clear now that time is the quantity common to all rates of change and it is common to standardise all rate measurements so that they represent a quantity per second. If you have studied calculus in mathematics, you will know that the derivative of a quantity with respect to time is the rate of change of that quantity.

Speed is the rate of change of distance or by how much the distance is changing each second. The average walker has an average speed of about 5 km h-1 which means that a distance of 5 km can be covered for every hour of walking. This converts to a speed of 1.4 metres per second (ms-1) which is the unit used more commonly in Physics to describe speed.

If we were to consider a 100 m sprinter running a race in a time of 15 s, we could easily calculate their average speed by taking the distance and dividing by the time taken. This provides an average speed for the sprint of 6.7 ms-1. The units are metres per second because we used metres in the numerator or our calculation and seconds in the denominator. We could easily have used km and hours in which case the calculation would have been 0.1 km divided by 0.0042 h and our answer would have been 24 km h-1. Suffice to say that 6.7 ms-1 and 24 km h-1are just two different ways to express the same speed.

In 84A1 - Speed, you will learn how to calculate speed using distance and time, the
difference between instantaneous and average speed and how to change the
subject of a three-variable equation.

We have calculated the average speed here because the sprinter's speed would have changed during the race. Initially their speed would have been zero and then increased to a maximum of 13.4 ms-1 at the end of the race. Between the start and the finish of the race, the sprinter's speed could have had any value between 0 ms-1 and
13.4 ms-1. This value for speed at any instant in time is known as the instantaneous speed. It is difficult to calculate the instantaneous speed for a journey where the speed changes without using graphs or differentiation so we more often work with average speed instead.

In the above example, the average speed can be calculated in another way. To calculate any average we add each of the measurements we have and divide by the total number of measurements. We have two measurements for our sprinter, a initial or starting speed (0 ms-1) and a final or end speed (13.4 ms-1). To calculate the average speed we add them together and divide by two to get (0 + 13.4) / 2 = 6.7 ms-1.

To see an example of how to do these calculations as well as the difference between instantaneous and average speed, make sure you watch the video in this section.

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84A2 - Units and Measurement

In this section you will learn to:

Describe quantities using symbols and units as prescribed by the SI, and

Use prefixes with units to indicate their order of magnitude.

International System of Units

To describe accurately how fast a car is moving along a straight road, we need a system of measurement. In Physics we use an international metric system known as the International System of Units (abbreviated SI from the French: Le Système international d'unités). This system has seven basic units or quantities and all other units are derived from them. Three of the most commonly used base quantities are length, mass and time while some common derived units include force, velocity and acceleration.



The international prototype of
the kilogram at BIPM in Paris.

Each quantity in the SI is uniquely defined by a common symbol and a unit. For example, the common symbol for time is t and its SI unit is the second (s). Similarly, the common symbol for mass is m and its SI unit is the kilogram (kg).

Each of the seven base quantities in the SI are defined very precisely using some common and reliable standard. A standard is something that does not vary so is the same all around the world. The best example is the common standard for mass, the kilogram. The international prototype of the kilogram (IPK) is the object kept and maintained by the Bureau International des Poids et Mesures (BIPM) in Paris. The standard kilogram was sanctioned in 1889 and takes the form of a cylinder with diameter and height of about 39 mm. It is made of an alloy of 90 % platinum and 10 % iridium and has a mass defined as one kilogram. The IPK has been conserved at the BIPM since 1889, initially with two official copies. Over the years, one official copy was replaced and four have been added. Access to the IPK and its official copies is under strict supervision of the International Committee for Weights and Measures (CIPM). The unit of mass is disseminated throughout the world by comparisons with the IPK made indirectly through a hierarchical system. The first step of these comparisons is normally with a subset of the "official copies" of the IPK, followed by calibrations of additional copies known as the "national prototypes" which are the primary national standards.

The symbols, SI units and standard for each of the seven base quantities are outlined in the table below.

Quantity Common Symbol Unit Name Unit Symbol Standard
length l metres m The distance travelled by light in vacuum in 1/299792458 seconds.
mass m kilograms kg The mass of the International Prototype Kilogram.
time t seconds s The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
electric
current
I amperes A The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newtons per metre of length.
temperature T kelvin K The fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
amount of substance N moles mol The amount of substance of a system which contains as many particles as there are atoms in 0.012 kilogram of carbon-12.
luminous intensity Iv candela cd The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Orders of Magnitude

In Physics we use orders of magnitude to describe how large or small numbers are. An order of magnitude is the scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, we use factors of ten and scientific notation to describe orders of magnitude. For example, the number 4.5 x 106 has order of magnitude six (from the power of ten) while the number 5.4 x 109 has order of magnitude nine. We say that 5.4 x 109 his three orders of magnitude (or 103 or 1000 times) larger than 4.5 x 106.

Some prefixes are used with certain units to describe the order of magnitude and avoid having to use scientific notation. For example, 9.6 x 10-6 s could be written as 9.6 µs (microseconds) because there are 1 x 106 microseconds in a second. You need to be familiar with and able to use the prefixes summarised in the table below when using units.

Name Prefix Symbol Example Decimal Power Ten Order of Magnitude Convert by multiplying base unit by ...
trillionth pico p picometres (pm) 0.000000000001 x 10-12 -12 0.000000000001 or 10-12
billionth nano n nanometres (nm) 0.000000001 x 10-9 -9 0.000000001 or 10-9
millionth micro µ micrometres (µm) 0.000001 x 10-6 -6 0.000001 or 10-6
thousanth milli m millimetres (mm) 0.001 x 10-3 -3 0.001 or 10-3
one - - metres (m) 1 x 100 0 -
thousand kilo k kilometres (km) 1 000 x 103 3 1 000 or 103
million mega M megametres (Mm) 1 000 000 x 106 6 1 000 000 or 106
billion giga G gigametres (Gm) 1 000 000 000 x 109 9 1 000 000 000 or 109
trillion tera T terametres (Tm) 1 000 000 000 000 x 1012 12 1 000 000 000 000 or 1012

84A3 - Displacement, Velocity and Acceleration

Here you will learn to:

Identify the vector quantities used to describe and measure motion by their symbols and units,

Distinguish between distance and displacement,

Describe velocity as the rate of change of displacement,

Use an equation to calculate either velocity, displacement or time for objects moving in one or two dimensions,

Distinguish between average and instantaneous velocity,

Describe acceleration as the rate of change of velocity, and

Use an equation to calculate either acceleration, velocity or time for objects moving in one or two dimensions.

Displacement

Some quantities have both a scalar and a vector version and length is one of these. The length of a journey in a car can be measured using the scalar we call distance, or using the vector equivalent we call displacement. For the purposes of studying physics, the vector quantity is usually the better choice as it tells us more about the particular quantity we have measured.

Quite simply, displacement is the shortest distance between two points, usually the start and end points of a moving object. When things move in only one direction and in one dimension (forwards only along a straight line), the distance and displacement have the same magnitude. We would just describe the displacement using a direction as well as a magnitude. If a car moved forwards into a driveway for 5.0 m and then reversed backwards by 2.0 m, the displacement of the car would be 3.0 m from the starting point. However, the distance travelled by the car would be
5.0 + 2.0 = 7.0 m.

Video 84A3 - Displacement explains the difference between distance and
displacement and how to convert between the two.

To use another example, if a car travelled west for 800 m on a straight road and then turned left and drove south for another 800 m, the distance travelled would be different to the displacement. The total distance travelled by the car would have been 800 + 800 = 1600 m but the displacement would have been 1131 m south-west. Make sure that you watch the video in this section to see more detail on these two examples and how to calculate displacement.

In some examples you will hear the word position used interchangeably with displacement. Position just means the location of the object in reference to its starting point or how far in a straight line it is from its starting point.

The symbol for displacement is usually s and its units are metres (m), although sometimes you will be expected to convert from km or other units such as cm. The symbol x is also acceptable for displacement and usually used when position is used in place of displacement.

Velocity

In the same way that average speed is the rate of change of distance, average velocity is the rate of change of displacement. This is similar to its speed but with a direction of motion. Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 kmh-1, its speed has been specified. However, if the car is said to move at 60 kmh-1 to the north, its velocity has now been specified. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path (the object's path does not curve). Thus, a constant velocity means motion in a straight line at a constant speed. If there is a change in speed, direction, or both, then the object is said to have a changing velocity and is undergoing an acceleration. For example, a car moving at a constant 20 kmh-1 in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.

Velocity is a vector physical quantity and both magnitude and direction are required to define it. The magnitude of velocity is called speed, a quantity that is measured in metres per second (ms-1) when using the SI (metric) system. For example, 5 ms-1 is a scalar (not a vector), whereas 5 ms-1 east is a vector. The symbol for velocity is v, although sometimes v is used for the final velocity and u is used for initial velocity.

Acceleration

In physics, acceleration is the rate at which the velocity of a body changes with time. Said another way, if the velocity of an object is changing (increasing or decreasing), it is accelerating. Acceleration is another vector quantity with magnitude and direction. It is always caused by a net force, as described by Newton's Second Law.

Mr Andersen from Bozeman Science explains velocity and acceleration and how to
solve problems involving displacement, velocity, acceleration and time.

Acceleration tells us by how much an object's velocity is changing each second. For this reason the units of acceleration are metres per second (velocity) per second. We write these units as m/s/s, m/s2 or best as ms-2. The symbol for acceleration is just a.

By way of example, an object such as a car that starts from a standstill travels in a straight line at increasing velocity. Because it's velocity is increasing, it is also accelerating in the direction of travel. When velocity and acceleration are in the same direction, objects are getting faster. If the driver were to put her foot on the brake pedal, the car would accelerate in the opposite direction to the velocity and the magnitude of the velocity would decrease until it came to rest. When velocity and acceleration are in opposite directions, objects are slowing down. For motion in a straight line, it is common to use positive and negative values to indicate direction. If the forward direction is positive, the reverse direction will be negative.

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84A4 - The SUVAT Equations

In this section you will learn to:

Use equations to calculate aspects of an object's motion when the acceleration is constant.

SUVAT Equations

SUVAT Equations

When acceleration is constant, there are three very useful equations that will help you in solving just about any problem for an object moving in a straight line or one dimension. They are called the SUVAT equations because the quantities s, u, v, a and t are used in the equations, with four of the symbols used in each equation.

Typically, problems are presented where you are given all but one variable and this is the one you must determine. Take for example a runner starting from rest and running an unknown distance in 90 s with a constant acceleration of 0.25 ms-2. We know that u=0 ms-1, t=90 s and a=0.25 ms-2. We could use v=u+at and solve for v to determine the final velocity. Then we could use s=ut+½at2 to determine the displacement of the runner.

Video 84A4 - Solving problems using the SUVAT equations.

These equations can only be used when the acceleration of a moving object is constant. That means that the object must be moving in a straight line or in one dimension. Examples of such problems include objects falling or rising under the force of gravity (the acceleration due to gravity is a constant 9.8 ms-2) and vehicles with velocity increasing or decreasing at a constant rate travelling in a straight line. A car that approaches a corner with a speed of 25 kmh-1 and leaves the corner with a speed of 35 kmh-1 is not accelerating at a constant rate. This is because it is not moving in a straight line and the direction of its velocity is changing. For this reason, the SUVAT equations could not be used in this instance.

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84A5 - Graphing Motion

Here you will learn to:

Draw and interpret:

- distance-time graphs,
- displacement-time graphs, and
- velocity-time graphs.

Because motion is most commonly described with respect to time, graphs with time on the x-axis and either distance, displacement or velocity on the y-axis are useful in describing motion. Such graphs are rich with information about the motion of an object and you should become familiar with their interpretation as well as how to draw them.

Distance-Time Graphs

Distance-time graph

A distance-time graph has distance on the y-axis and time on the x-axis. Since distance is a scalar, the y-axis can only ever be positive. A negative value would indicate that is was displacement and not the distance travelled. Because speed is the rate of change of distance with respect to time, the gradient of a distance-time graph is the speed. Where the graph is represented by a straight line, the speed is constant. If the graph is curved, then the speed is changing and the object is accelerating.

To summarise the features of a distance-time graph:

  • The gradient of the graph is speed,
  • A straight line means constant speed, and
  • A curve means that the speed is changing and the object is accelerating.

Displacement-Time Graphs

Displacement-time graphs have distance on the y-axis and time on the x-axis. Displacement is a vector so the y-axis can have both positive and negative values. Velocity is the rate of change of displacement with respect to time so the gradient of a displacement-time graph is the velocity. Where the graph is represented by a straight line, the velocity is constant. If the graph is curved, then the velocity is changing and the object is accelerating. If a tangent were drawn on the curved part of a displacement-time graph and its gradient calculated, this would show the instantaneous velocity at that point (remember that the velocity is changing when it's a curve). When the graph has a straight line, the velocity is constant so the instantaneous velocity does not change.

Displacement-Time Graph

If the gradient of the graph slopes toward the right, the gradient is positive, the velocity is also positive and in the same direction as the positive displacement. Conversely, if it slopes to the left the velocity will be negative and in the opposite direction to the positive displacement.

To summarise the features of a displacement-time graph:

  • The gradient of the graph is velocity,
  • A straight line means constant velocity,
  • A curve means that the velocity is changing and the object is accelerating,
  • A positive gradient (sloping to the right) means positive velocity, and
  • A negative gradient (sloping to the left) means negative velocity.

Velocity-Time Graphs

Velocity-time graphs have velocity on the y-axis and time on the x-axis. Velocity is a vector so the y-axis can be positive and negative which means that the graphs can appear above and below the x-axis. Since acceleration is the rate of change of velocity, the gradient of a velocity-time graph represents the acceleration of the object. Where the graph is represented by a straight line, the acceleration is constant. If the graph is curved, then the acceleration is changing. If a tangent were drawn on the curved part of a velocity-time graph and its gradient calculated, this would show the instantaneous acceleration at that point. When the graph has a straight line, the acceleration is constant so the instantaneous velocity does not change.

If the gradient of the graph slopes toward the right, the gradient is positive, the acceleration is also positive and in the same direction as the positive velocity. Conversely, if it slopes to the left the acceleration will be negative and in the opposite direction to the positive velocity.

Velcoity-Time Graph

You will remember that acceleration is given by the equation, v=Δs/Δt. If we rearrange the equation to make displacement the subject of the equation we get, Δs=v.Δt. This tells us that the displacement is the product of velocity and time. On a graph of velocity and time, the product of these two quantities is the area under the graph and for that reason displacement is given by the area under a velocity-time graph.

Since displacement is a vector, it can be both positive and negative. On a graph an area above the x-axis would represent a positive displacement and the area below the axis a negative displacement.

To summarise the features of a velocity-time graph:

  • The gradient of the graph is acceleration,
  • A straight line means constant acceleration,
  • A curve means that the acceleration is changing,
  • A positive gradient (sloping to the right) means positive acceleration,
  • A negative gradient (sloping to the left) means negative acceleration,
  • The area under the graph is the displacement,
  • An area above the x-axis is positive displacement, and
  • An area under the x-axis is negative displacement

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84A6 - Uniform Circular Motion

Uniform circular motion is circular motion with a constant orbital speed. It is different to the motion we have seen above because it is occurring in two dimensions as the object in question moves in a circle. Even though the speed of the object is constant, the velocity is changing because the direction is changing. Because the velocity is changing, it must be accelerating.

As an example of circular motion, imagine you have a rock tied to a string and are whirling it around above your head in a horizontal plane. If you were to let go of the string, the rock would fly off at a tangent to the circle - a demonstration of Newton's First Law of Motion. He said that an object would continue in uniform motion in a straight line unless acted upon by a net force. In the case of our rock, the force making it accelerate and keeping it within a circular path is the tension in the string and it is always directed back towards the hand at the centre of the circle. Without that force the rock would travel in a straight line. So, if the magnitude of the velocity is constant but its direction is changing, the velocity must be changing and it is, therefore, accelerating. A net force is required to accelerate the rock and as stated above, this force is the tension in the string. The force needed to keep an object moving in a circle is also called centripetal force and the acceleration is called the centripetal acceleration.

The same is true of a spacecraft in orbit around the Earth or any object in circular motion - some force is needed to keep it moving in a circle or accelerate it and that force is directed towards the centre of the circle. In the case of the spacecraft, it is the gravitational attraction between the Earth and the spacecraft that acts to maintain the circular motion and keep it in orbit. When a car turns a corner at a constant speed, it is accelerating because it is changing direction. The force causing the acceleration is the friction between the types and the road as the driver turns the wheel.

The diagram on the left allows us to work out the direction of the centripetal acceleration - which must also be in the direction of the centripetal force. The object is shown moving between two points A and B on a horizontal circle. Its velocity has changed from v1 to v2. The magnitude of the velocity is always the same, but the direction has changed. Since velocities are vector quantities we need to use vector mathematics to work out the average change in velocity. You will see how to do this in the next section of work. In this example, the direction of the average change in velocity is towards the centre of the circle. This is always the case and thus true for instantaneous acceleration.

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