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KS4 Forces and motion

It is surprising that the AQA Specification should place Forces and motion so far down in the list of sections in its Forces chapter, but if you are working through the Forces chapter in order, then you will already already have worked through Forces and their interactions,
Work done and Energy transfer,
Forces and elasticity,
Moments, levers and gears

and Pressure and pressure differences in fluids.

So now, at last, we come to this very important Forces and motion section.

4.5.6.1 Describing Motion

When a resultant force acts on an object, the object will either speed up, slow down or change direction.

On the other hand, if there is NO resultant force acting on the object, then the object might still be moving but at a constant speed.

In this section we need to learn how to describe:
Slowing down
Speeding up
Changing direction

To do this we will need to learn some new "terminology" (some new words) such as DISTANCE, DISPLACEMENT, SPEED and VELOCITY.

1. Distance and Diplacement

Let's start with the easy one of these two:

A. Distance is how far an object moves.

So, if an athlete runs 100m in a straight line, the "Distance" is 100m; if another athlete runs all the way round the oval track then he/she will have run a distance of 400m.

In either case, the fact that one athlete ran in a straight line whilst the other ran in an oval (and ended up back at the start) was irrelevant as far as Distance was concerned.

So, what we are saying is, is that Distance takes no account of the direction travelled; Distance is a SCALAR quantity (you should remember this from the Forces page.)

For a Distance measurement the only thing that matters is how far the object has moved and we measure it in metres, m.

Which way to go?
The DISTANCE is the same no matter which Direction he chooses.

These Distances are absolutely identical.
Direction of travel is irrelevant as far as Distance is concerned.

So, what then is Displacement?


B. Displacement is how far an object moves in a specified direction.

Or, Displacement is Distance in a specified direction.

An example of a Displacement quantity would be, "The boy walked 2Km north."

Or, the woman rode her motorcycle for 50Km west out of the city.

In both examples it is the addition of the direction that changes the quantity from a basic Distance type to a Displacement type.

Since Displacement is a quantity that contains two pieces of information, one regarding the distance travelled and one regarding the direction of travel, it is a VECTOR quantity.

Nevertheless, we still use the unit metres,m, so apart from the need to specify the direction of travel it is not hugely different to Distance. In fact, whilst examples of Distance can be of objects moving in ovals, circles, spirals etc, the need to specify the direction with Displacement means that its examples are ALWAYS of objects moving in STRAIGHT LINES.

The Displacements are different.
10Km West is not the same as 10Km East.

These are 3 totally different Displacements.
In fact, in the curved example, the final displacement is much less than 100m; can you see this? Despite the curve, the overall displacement is just the straight line from the start to the finish.

2. Speed and Velocity

Once again, let's start with the easy one of these two:

A. Speed: There are a number of ways in which we can describe the meaning of the term "speed".

One is: Speed is how much distance is covered in how much time.

Another is: Speed is the rate of movement.

But perhaps the clearest is: Speed is the distance travelled divided by the time taken.

This final definition also leads us straight into the equation (or formula) for Speed, which is:

KS4 students must be able to re-arrange the Speed equation in order to use it to calculate the distance travelled !

After rearranging the equation to calcuate distance travelled, we arrive at:

OK, let's do another example calculation, but this time we will calculate a distance travelled.

Relative Motion

Have you ever sat on a train and looked out of a window at a train next to you and then after many seconds realised that your train was moving, but so was the other one! Both trains moved off at the same time and at the same speed and in the same direction such that it appeared to you that both were stationary.
Eventually your train picks up speed and slowly pulls away from the other train; relative to the other train your train is moving slowly even though relative to the land it might be moving very fast.

We call this "relative motion". We also notice it when we (as the observer) look across from one car to another when pulling off from traffic lights; our car appears stationary "relative to" the nearby car.
But when we drive towards and past an approaching car it seems that we pass each other at a high speed; in this situation, the speed "relative to" an observer in one of the cars is the sum of the individual speeds (this is why head on collisions of vehicles are the worst and often deadly).

Typical Personal Speed Values:

  1. Walking 1.5m/s
  2. Running 3m/s
  3. Cycling 6m/s

All of these values will vary with age and fitness of the person and the state of the terrain, eg hilly or muddy compared to smooth, level tarmac. Also, it is obvious that a person's speed varies during a walk or a run or a ride, so these values are only average values. NB. When speed varies over a period of time, we call that period of motion, "non-uniform motion".

Other forms of transportation can have much higher average speeds eg a car could have a speed between 13 and 31m/s, whilst a jet airplane can cruise at over 200 m/s.

Sound travels faster than all of these things. It has a typical speed of 330m/s in air (it travels even faster through water and faster still through solids! You can find out those values yourself.)

One last thing to say about Speed.
Speed, like distance, conveys only one bit of information. It gives us just a value, a size, a number. So, like distance, Speed is also a SCALAR quantity.

If one person runs West at 4m/s whilst another person runs East at 4m/s, they have exactly the SAME speeds. Speed ignores direction.

Now let's consider VELOCITY.

B. Velocity might be a less familiar word but now that we understand Speed, we can easily gain an understanding of Velocity because Velocity is simply "Speed in a specified direction."

So, If one person runs West at 4m/s whilst another person runs East at 4m/s, they do NOT have the same VELOCITIES.

Their Speeds are the same because "speed" takes no account of their direction; but their Velocities are totally different because their directions are totally different.

Since Velocity, like Displacement, conveys 2 pieces of information, a size and a direction, it is a VECTOR quantity.

Calculating Velocity, however, is exactly the same as calculating Speed; you use the very same equation, though at KS4 you need to know symbols for the various terms, so here they are:

Speed, Velocity and Circular Motion

An object moving in a circular path at constant speed is NOT moving at constant velocity !

Why is this?

It is because, as the object moves along its circular path, eg a car going round a roundabout, its direction keeps changing.

So, although the size part of its velocity might be constant, the direction part is not, hence its Velocity is NOT constant.

This applies even if the object is moving in part of a circle eg a car turning left or right at a junction.

Examples of Circular Motion:
Cars moving around a roundabout or part of a roundabout
A planet orbiting a star eg the Earth orbiting the Sun
A satellite orbiting a planet
A weight being swung around on a string, rope or cable eg as with an Olympic "hammer" thrower.

I'm sure you can think of lots more. Here's one:

Since the two carriages are fixed to the same ferris wheel they move at the same SPEED, always.
But, because they are ALWAYS moving in different DIRECTIONS, their VELOCITIES are ALWAYS different
and because their DIRECTIONS keep changing, their VELOCITIES keep changing.

OK. We have completed what we set out to do - to introduce some new terminology, the words Distance, Displacement, Speed and Velocity. We are now ready to take the next step in Describing Motion.

4.5.6.1.4 The distance-time relationship

Describing Motion using Distance-Time Graphs

Distance-time graphs such as the one shown below can be used to describe the motion of an object travelling in a straight line and in particular to calculate the speed of an object at various points on its journey.

Since Speed = Distance / Time , the Gradient of the line on the Distance-time graph reveals the speed of the object.

The graph above is the simplest Distance-time graph since it shows just a single slope (or gradient) line. This will tell us about just one speed of the object.

Let's calculate the speed of the object revealed by this graph.

More complex graphs will have multiple slopes and can tell us how the speed of an object changes during its journey.

The second graph, below, is a more complex one showing multiple slopes which will reveal multiple speed values.

Graph 2: Three slopes (including the flat section) reveal three speeds.

Let's consider another graph. Look at the 3rd Distance-time graph.

Graph 3: Four slopes reveal four speed values.

The main difference compared to the first two graphs is that it has a section with a negative slope. What does this signify? It simply means that the object has turned around and is heading back towards its starting point (the distance is getting less as time increases, reaching 0m, the starting point, at 100s).

So, the calculations for graph 3 are the same as for graph 2 but with one extra section:

So far, so good. But what would the Distance-time graph mean if the "lines" were not straight but were curved?

Look at the fourth graph. It has one section but it is curved.

Its slope or gradient gets steeper and steeper as time increases. So, this object is getting faster and faster as time increases. The object is speeding up; it is accelerating.

So a curved "line" on a Distance-time graph means that an object is accelerating.

We can reveal one more fact about curved Distance-time graphs, but this is quite a tricky concept so you may have to read it a few times:

If we draw a tangent to the curve at a time point then by finding the gradient of that tangent we can work out the speed of the object at that time point. (If you don't know what a "tangent" is, see the blue left side boxes.)

For example, see the fifth graph.

A tangent to the curve has been drawn at the time point of 50s.
The gradient of this tangent is: 10m / 80s or 0.125 m/s which is the speed of the object at the time of 50s

If we wanted to find the speed at the time of say, 80s, we would simply draw a new tangent at the 80s point and re-calculate the gradient. We would expect the speed to be higher because the gradient would be steeper.

4.5.6.1.5 Acceleration

Another term relevant to "describing the motion of an object" is acceleration.

As we stated above, when an object "speeds up", it is accelerating

The acceleration of an object can be defined as the rate at which its speed is increasing or changing; or how quickly its speed increases or decreases.

The greater the change of speed or the smaller the time taken for the change to take place, the larger the acceleration of the object.

This leads to an equation or formula for acceleration:

Describing Motion using Velocity-Time Graphs

Using a Distance-Time Graph to describe the motion of an accelerating object is possible; we saw that above, but it involved curved lines which makes such graphs difficult to draw and to analyse.

A much easier graph to use to describe the motion of an accelerating object is a Velocity- Time Graph such as graph 1 shown below.

The gradient of this line is:
change in velocity, over, time taken. Which is the acceleration.

So, the gradient of a Velocity-Time graph reveals the Acceleration of the object!

At first glance it looks identical to a Distance-Time graph, so it is VITAL that you look carefully at the y-axis to see that it is, in this case, labelled with the word "Velocity".
(The meaning of the lines on a Velocity-Time graph is totally different to the meaning of the lines on a Distance-Time graph ! You have been warned.)

So whilst the gradient of a line on a DISTANCE-Time graph tells you its Velocity,
the gradient of a line on a VELOCITY-Time graph tells you its Acceleration.

Graph 2 describes a more interesting journey!

Here there are 3 sections, so 3 different acceleration values. We can calculate each of them:

There is one more thing to say about Velocity-Time Graphs.
They can be used to work out the Distance Travelled by a moving object.

The area below the line of a Velocity-Time graph represents the Distance Travelled.

Now, if you are reading this then I know you are an inquisitive person and you are wondering how can it be that the area below the line of a velocity-time graph gives us the distance travelled?

The justification is simple:
We know that Distance travelled= Velocity x Time
and if you look at the axes of the graph, you should notice that "velocity x time" is the same as saying "height x width" or "area of the graph" or more precisely "the area below the line". So, that is why:

Distance Travelled = Area below the line of a Velocity-Time graph

Note: the word "Displacement" can be used for "Distance" here and a Velocity-Time graph could show a line below the x-axis which would mean that the object had reversed direction.

Finding Acceleration Without Knowing the Time Taken !

Consider the following question:
A cyclist is cycling at 4 m/s but then speeds up to 6 m/s in a distance of 40 m; what was its acceleration?

It would seem that you couldn't answer the question because you are not given the "Time taken" which you need in order to use the formula:
acceleration = change in velocity / time taken

But "distance" and "time" are always interconnected due to the relationship,
velocity = distance / time
so, without going into the detail, it turns out that you can derive an equation that includes acceleration, velocity and distance but doesn't need to use time.

This is the equation, first in words:

(final velocity)2 - (initial velocity)2 = 2 x acceleration x distance

And here it is using symbols:

v2 - u2 = 2 a s
where v is final velocity in m/s
u is initial velocity in m/s
a is acceleration in m/s2
and s is distance in m.

Luckily this equation is not one that you will have to remember; unlike many of the others we have mentioned, it will be given on an equation sheet in an examination.

Anyway, let's now use it to solve the problem that we presented at the start of this section:

Would an accelerating object, in our real world, keep getting faster and faster and faster? Or, is there a limit to how fast an object can go?

Velocity-Time Graphs and Terminal Velocity

This next section straddles between the work we are currently doing on "motion" eg velocity-time graphs, and the work that we did in a previous section on "Forces and their interactions".

Terminal Velocity - what does the phrase mean?

The phrase literally means "end or final velocity"

It is used in situations where an object can "go no faster".

One of the best examples of this is that of a skydiver who jumps from an aeroplane and begins to fall, faster and faster.
But, what we find is that the skydiver quite quickly (after about 10s) reaches a velocity beyond which he doesn't go; so he has reached a "terminal velocity" and carries on falling at this velocity (which is still fast, about 60 m/s or over 120 mph) until he opens the parachute and then slows down dramatically before he lands.

Ok. We need to do 2 things concerning Terminal Velocity
1. Draw a velocity - time graph to illustrate it, and
2. Explain it using your knowledge of Forces (so you will have to have read all the previous sections on Forces).

1.So, look at the graph below.

It shows the velocity of a skydiver in the first 20 seconds after jumping from an aeroplane. We need to examine it carefully:

The diver jumps out at time zero and accelerates at a high rate towards the earth; we see this as a very steep section between 0 and 1 second.

But even by just 1 second we notice that the "line" has begun to curve meaning that the size of the acceleration is getting less and less. By 4 seconds the gradient of the line is much less than at the start so although the diver is still going faster, his acceleration is reduced.

As 10 seconds passes, the diver's velocity begins to level out (his acceleration is becoming zero) and the diver is reaching his Terminal Velocity. He finds that he just can't go any faster. This is illustrated on the graph with the line becoming horizontal at 60 m/s which is the terminal velocity for this diver.

Finally, what is happening at 16 and 18 seconds?
At 16 seconds the diver opens his parachute so he slows down rapidly to just 10 m/s reaching a new, much lower, terminal velocity.
He then stays at this new terminal velocity until he lands at which point his velocity will become 0 m/s (we haven't shown this on the graph).

2. To explain Terminal Velocity for the skydiver using our knowledge of forces, look at the first diagram of the skydiver below.

This shows the skydiver having just jumped out of the aeroplane, so there is very little air resistance opposing his weight.

So, the Resultant Force (remember: the single force that replaces all others) is:
Resultant force = Weight - Air Resistance
which will be a large downward force, causing a large downwards acceleration, so the diver falls faster and faster. (This is the steep part of the Velocity-Time graph.)

However, since the diver is now falling faster, there will be more air resistance, see diagram 2.

Now, the Resultant force (Weight - Air Resistance) is less than it was at the start, so although there is still a resultant force downwards causing the diver to still fall faster, his acceleration decreases. This is shown on the graph as the curving section at about 4 seconds.

Since the diver is going even faster, the air resistance has become even larger. This is shown in the final diagram, number 3.

What we notice is that the Air Resistance is now EQUAL to the Weight, so

Resultant Force = Weight - Air Resistance
                        = ZERO

So, if the Resultant force on the diver is zero newtons the diver neither speeds up or slows down; instead he or she stays at a constant speed. This is the horizontal line section on the graph and occurs at about 10 or 11 seconds. The diver has reached Terminal Velocity.

The diver would fall at this velocity, about 60m/s, until he/she hits the ground if the parachute isn't opened. So, what happens when the parachute opens? Does the upward Air Resistance become larger than the downward Weight?

Well, yes and no!

Yes, the huge surface area of the parachute produces a very large upward force of Air Resistance and due to the initial high falling speed of the diver it is, at first, much larger than the weight but only for a few seconds; just long enough for the velocity of the skydiver to reduce quickly causing the Air Resistance to decrease until it eventually equals the Weight again. So, the diver arrives at a new, smaller Terminal Velocity.

Notice, that in all of this, the only force that never changes is the diver's Weight.

Before we move back to a more pure study of forces, you need to have a go at some questions on the Advanced acceleration equation and on Terminal Velocity.

4.5.6.2 Newton's Laws of motion

In our first exploration of the world of Forces (Forces and their interactions) we encountered the great man, Isaac Newton, finding that we name the unit of force, the "newton N", in honour of him. Then we met him when we studied Weight and encountered his "Universal Law of Gravity".
Now we are going to meet him again as we study his 3 Laws of Motion.

Newton realised that all objects obey these 3 laws and even today when space scientists design rockets to send satellites into space or probes to distant planets, it is to Newton's 3 laws that they turn in order to make their calculations in order to send their expensive cargo safely into space. Today, or in the future, we can predict all the planetary movements because their motions conform to Newton's Three Laws of Motion!

Now, the good news is that if you have worked through all our "Forces" content, then all of what follows should be easy!

4.5.6.2.1 Newton's First Law

"If the resultant force acting on an object is zero and the object is stationary, the object remains stationary.
If the resultant force acting on an object is zero and the object is moving, the object continues to move at the same speed and in the same direction. (ie. continues to move at the same velocity.)"

In Forces and their interactions, we described this situation as a state of "Equilibrium" and the forces that caused a state of Equilibrium are known as "Balanced Forces".

The way we summarised the effect of balanced forces was to say "balanced forces cause NO CHANGE to the motion of an object".

So, if the object was stationary before the balanced forces acted, it remains stationary after they acted.

Here's an example:

Before any forces are applied, the object is stationary.

Now, forces are applied, but since they are BALANCED, the object remains in its state of EQUILIBRIUM and its motion does NOT change. It remains stationary.

Also, if an object was moving at constant velocity before the balanced forces acted, it remains moving at constant velocity after they acted.

Here's an example:

This car is currently moving at a constant speed; (like the stationary box, it is in a state of EQUILIBRIUM).

Now, the engine produces an extra 7N of forward force, but there is also an additional 7N of air resistance. The resultant of these two forces is zero, so these forces BALANCE so the car remains in EQUILIBRIUM. Its motion does NOT change. It ontinues at its constant speed of 2m/s.

So, Balanced forces or a Resultant Force of Zero, cause NO CHANGE to the motion of an object. That's it!

Knowing Newton's 1st Law allows you to deduce things, like Sherlock.

A bit like the way Sherlock Holmes can deduce facts from pieces of knowledge, so we can make deductions based on our knowledge of Newton's 1st law.

For example:
If we are told that an object (a car, a bus, a boat etc) is moving at constant velocity, then we can deduce that the forces acting on it are balanced or that the resultant of the forces acting on it is zero.

If we are told that an object is stationary then we can deduce.........as above.

If we are told that the driving force of a boat is 500N but that the boat is moving at a constant velocity, then we can deduce without hesitation that there must be an opposing resitive force of 500N balancing this forward force, producing a resultant force of zero.

We can say this because we are told that the object is moving at constant velocity.

If the driving force is increased causing the boat to speed up, but then eventually settles to a higher constant velocity, we can deduce that the resistive force must also have increased to exactly balance out the driving force.

One last thing..

As we have realised, Newton's 1st law "paints a picture" of objects in a state of equilibrium, that is, either stationary or moving at a nice steady speed in a nice straight line. That is the state of most objects if allowed to move freely in space and even here on Earth objects will try to move in straight lines and stay at constant speeds if they can (and they would if it wasn't for the resistive effects of air resistance and friction that slow things down).
Newton noticed all of this and summed it all up by saying that objects possessed a property that he called "Inertia", which he described as an object's reluctance to be CHANGED from its current motion.
Large, heavy object are said to have a large Inertia because it is hard to CHANGE their motion eg to stop them or slow them down or change their direction etc. Small, light objects are said to have a small Inertia because it is easy to CHANGE their motion eg to speed them up etc.

Obviously, I hope you agree, the Inertia possessed by an object is equivalent to its mass.
So, Inertia is also known as Inertial Mass and like mass, it is measured in kg.

Inertia! A new word, learn it.

4.5.6.2.2 Newton's Second Law

Newton's 2nd law is very quick and easy to write:
"The acceleration of an object is proportional to the resultant force acting on the object, and inversely proportional to the mass of the object."

We can split it into two statements if that helps:
1) The acceleration of an object is proportional to the resultant force acting on the object, and
2) The acceleration of an object is inversely proportional to the mass of the object.

The first statement is pretty obvious, don't you think? The larger the resultant force that you apply to an object the greater will be its acceleration, eg the harder you kick a ball the greater will be its acceleration, yes!

Another way of expressing the first statement is using symbols:

a ∝ F

where a = acceleration, F= resultant force and ∝ means “proportional to”.

(Notice that the "F" is the Resultant force, not just "force". It is vital that you remember this and always work out the resultant force in any situation.)

The second statement is also not too difficult to grasp as long as you know what is meant by "inversely proportional". Here is an example - an object with a LARGER mass will accelerate LESS eg if you kick a brick with a certain force it will accelerate less than if you kick a football with the same force; why? Because, the brick has a LARGER mass.

LARGER mass, SMALLER acceleration
SMALLER mass, LARGER acceleration
This is the meaning of the "inversely proportional" relationship.

Using symbols, the second statement can be written:

a ∝ 1 / m
where m = mass of object

So whilst Newton's 1st law was all about Balanced forces or Resultant Forces of Zero, all of which cause a beautiful state of Equilibrium and no change to motion, his 2nd law is totally different; it is all about Unbalanced forces and Resultant Forces that are definitely not zero, causing change to motion and so to acceleration.

Newton's 2nd Law Calculations

Whilst Newton's 1st law led us to emulate Sherlock, deducing things here, there and everywhere, his 2nd law leads us down the Einstein route into the world of equations and calculations. We have had a hint of this already in the two small equations of his 2nd law.

But before we do proper calculations we can put the two equations together to make one equation:

Now we are ready to do a few example calculations.

Time for you to have a go at some "F = m a " questions. Remember to leave a space between your answer and the unit and due to the difficulty of typing the "2" above the s in the acceleration unit, just type m/s2 for now.

NB Before we leave Newton's 2nd law - do you remember the word "Inertia"?
We met it briefly in our discussion of Newtons's 1st law. We said that inertia was basically a measure of the mass of an object and so we called it "inertial mass".
Well, Newton's 2nd law gives us a neat way of calculating this inertial mass. All we have to do is rearrange our "F = m a " equation as follows:

m = F / a

Also, from Newton's 2nd law we have learnt that the acceleration of an object is inversely proportional to the "mass" of an object, so substituting the term "inertial mass" for "mass" we can say that:

The inertial mass of an object is inversely proportional to the acceleration of an object,

or

The inertial mass of an object is a measure of how difficult it is to make it accelerate or change velocity.

(This might take a while to get you head around, but keep reading it and thinking about it.)

4.5.6.2.3 Newton's Third Law

"Whenever TWO OBJECTS interact, the forces they exert on each other are always equal in size and opposite in direction."

Notice that Newton's 3rd law concerns TWO OBJECTS interacting.
It is not about ONE OBJECT which might have two or more forces acting on it, such as....

OK. Now that we have that clear, let's think of some examples where the 3rd law DOES apply.

Example 1. A person pushing a supermarket trolley

The person is Object 1 and the trolley is Object 2.
Object 1 pushes onto Object 2;
Object 2 pushes back onto Object 1 making the trolley move forwards.
The 2 forces are Equal in size but Opposite in direction and act on TWO different objects.

Example 2. A sprinter leaving his/her starting blocks

The person is Object 1 and the blocks are Object 2.
Object 1 pushes onto 2;
Object 2 pushes back onto 1 making the runner move forwards.
Just the same as in example 1.

We used to call the pairs of forces "Action" and "Reaction" forces; so, the first Object of the pair produces the Action force (the person pushing the trolley or the runner pushing the blocks) and the second Object of the pair produces the Reaction force (the trolley or the blocks). The nice thing about these "old fashioned" terms is that they gave rise to the easy way of stating Newton's 3rd law, which was "For every Action there is an equal but opposite reaction".

Newton's 3rd law explains how certain "obvious" things actually work!

For example, do you know how a boat propeller actually drives a boat forward?

Answer: The propeller pushes on the water; the water pushes back on the boat, driving it forward! Simple explanation.
The Propeller is Object 1 and the Water is Object 2.

Another example - do you know how a jet engine drives an aeroplane forward?

Answer: The jet engine pushes on a huge mass of air; the air pushes back on the aeroplane, driving it forward.
The engine is Object 1 and the mass of air is Object 2.

NOTICE, in each of the examples, the second object, Object 2 produces a force which always pushes BACK on the first object.

4.5.6.3 Forces and Braking

4.5.6.3.1 Overall Stopping Distance

When a driver of a car (or a bus, or a van, lorry etc) is driving along a road but then sees a hazard such as a person stepping onto the road, the driver knows that he or she has to begin to apply the vehicle brakes.
But before the brakes have even begun to be applied there is a time delay between the driver seeing the hazard and the brakes being applied.

This time delay is unavoidable and it is typically about 0.7s; it is known as the Reaction Time.
During this "Reaction Time", the vehicle is still moving at its previous speed and the distance it moves during this "Reaction Time" is called the Thinking Distance.
So, finally, after the Reaction Time, the driver starts to apply the brakes and the vehicles starts to slow down.
The distance it travels whilst the brakes are being applied, until it stops, is called the Braking Distance.

So, what is the total distance that the vehicle travels between first seeing the hazard and finally stopping?

The Overall Stopping Distance = Thinking Distance + Braking Distance

In other words, the overall stopping distance is simply the sum of the 2 distances:
1. The distance that the vehicle travels whilst the driver is still "thinking" or reacting, and
2. The distance that the vehicle travels whilst the driver is actually applying the brakes.

The driver has to hope that these 2 distances don't add up to a number that is too large for the hazard immediately at hand !

We will now consider these two distances seperately.

4.5.6.3.2 Reaction Time and Thinking Distance

Reaction time varies from person to person. For some people it can be as low as 0.2s, but such people are rare. For most people it is between 0.7s and 0.9s, but it can be even longer depending on other factors.

The larger the Reaction Time, the larger the Thinking Distance (and the greater chance of hitting the Hazard!)

So, it is important to know the Factors that affect Reaction Time.

Factors include: tiredness, alcohol, drugs (both illegal and prescribed) and distractions (passengers, music, phone including hands free, distractions from outside the vehicle).
So there are a lot of things that can affect a driver's reaction time and hence, the Thinking Distance.

But there is another factor that affects the Thinking Distance which is not directly affected by the reaction time and that is, obviously, the speed of the vehicle.
In other words, if the vehicle was driving slowly when the hazard is spotted then it wont travel far during the Reaction Time; but if the vehicle was driving fast when the hazard was spotted then it will travel far during the Reaction Time.

Thinking Distance = Speed of vehicle x Reaction Time (ie from Distance = Speed x Time)

See the small table on the right or below.
Check for yourself that the Distances are calculated correctly, and also notice that the Distance doubles as the speed doubles.

So Thinking Distance depends on Reaction Time and on Speed of Vehicle. It is "proportional" to each of them.

You can see why the Highway Code stresses the importance of driving below the speed limits at all times and for keeping alert at all times, taking rests on long journeys and avoiding distractions. In the future, your life and the lives of others could depend on you taking care. Imagine the consequences if you don't !

4.5.6.3.3 Factors affecting braking distance 1

The braking distance of a vehicle can be affected by:
a) adverse road and weather conditions (eg. if the road surface is in a poor state of repair or due to bad weather it is wet or icy)
and
b) poor condition of the vehicle (eg. its brakes are worn or not operating correctly, or the tyres are worn or damaged).

All of the factors mentioned above will adverely affect the Braking distance of a vehicle at any speed, ie. it will increase its Braking distance making it less likely that the vehicle will stop before the Hazard is reached!

This is why the governments of the UK have ruled that it is a legal requirement for all vehicles to pass an MOT test every year in order to be able to drive on the roads; amongst other things, this test checks the brakes and the tyres of the vehicle.

The following chart shows how the Braking Distance varies with the speed of the vehicle.

The braking distance doesn't just double as the speed of the vehicle doubles (which is how the Thinking Distance varied with vehicle speed).
Instead, if you look at the dark vertical line at the end of the 20mph "red" arrow and follow it down to the 40mph arrow, you will notice that the 40mph arrow has much more than doubled in length (from 12m to 36m).
Similarly, look at the black line at the end of the 30mph arrow and follow it down to the 60mph arrow and you will notice that it has much more than doubled in length (from 23m to 73m)

To illustrate this fact more dramatically, look at the illustration below; this is just the same red arrows, but rotated so that they can be turned into a graph!

As you can see, when a "line" is plotted, it is not a straight line but is a curve whose gradient is getting steeper and steeper as the speed of the vehicle increases.

So, Braking Distance is not proportional to speed, it is MORE than proportional to speed!

This is why a vehicle going just a little more than a speed limit is potentially VERY dangerous, not just a LITTLE bit dangerous.

4.5.6.3.4 Factors affecting braking distance 2

Have you noticed that since we started this piece of work on Forces and Braking, we have talked a lot about braking and about speed and distance and time, but we haven't mentioned FORCES at all ! That can't be right, can it? So, at last we are going to turn this section into a proper Physics topic.

This is what happens:

When a brake is applied, a friction force acting on the brake disc gradually reduces the kinetic energy of the vehicle, turning it into heat energy at the disc, slowing down the vehicle.

Another way of stating this is:
When a brake is applied, work is done by the friction force between the brakes and the wheel.
This work done reduces the kinetic energy of the vehicle, slowing it down, but also causing the temperature of the brakes to rise (so it does one good thing but also one less good thing).

The greater the speed of the vehicle the greater the braking force needed in order to slow it and stop it within a certain distance (this is pretty obvious, don't you think?).

On the other hand, if a greater braking force can be applied at a particular speed then a greater deceleration should be achieved, but it could also cause the brakes to overheat and if brakes overheat TOO much, they actually STOP working completely leading to loss of control !!! Yikes.
So it is wise not to drive too fast in the first place.

By using Newton's Second law of motion, we can estimate the forces involved in the deceleration of road vehicles.

Estimating braking forces

Let's get straight into an example:

Finally, to conclude this section, don't forget what we stated near the start of this section:

The Overall Stopping Distance = Thinking Distance + Braking Distance

Now its time for you to test your knowledge of these ideas. Don't forget to leave a space before any unit symbols and type m/s2 for m/s2.