MOVING ABOUT
Introduction
Mechanics
Motion
Motion Graphs
Relative Velocity
Force
Newton's Laws of Motion
Mass & Weight
Usefulness of Vector Diagrams
Vector Addition
Vector Subtraction
Vector Components
Tensions in Strings - Extension Topic
Circular Motion
Work
Energy - Kinetic, Potential, Transformation of Energy
Momentum
Conservation of Momentum
Conservation of Momentum During Collisions
Inertia
Practicals Page - lots of good pracs
Worksheets Page - Forces, Energy, Momentum & Vector Analysis
Useful Links Page
INTRODUCTION:
Increased access to transport is a feature of today’s society. Most people access some
form of transport for travel to and from school or work and for leisure outings at
weekends or on holidays. When describing journeys that they may have taken in
buses or trains, they usually do so in terms of time or their starting point and their
destination. When describing trips they may have taken in planes or cars, they
normally use the time it takes, distance covered or the speed of the vehicle as their
reference points. While distance, time and speed are fundamental to the understanding
of kinematics and dynamics, very few people consider a trip in terms of energy, force
or the momentum associated with the vehicle, even at low or moderate speeds.
The faster a vehicle is travelling, the further it will go before it is able to stop. Major
damage can be done to other vehicles and to the human body in collisions, even at
low speeds. This is because during a collision some or all of the vehicle’s kinetic
energy is dissipated through the vehicle and the object with which it collides. Further,
the materials from which vehicles are constructed do not deform or bend as easily as
the human body. Technological advances and systematic study of vehicle crashes
have increased understanding of the interactions involved, the potential resultant
damage and possible ways of reducing the effects of collisions. There are many safety
devices now installed in or on vehicles, including seat belts and air bags. Modern road
design takes into account ways in which vehicles can be forced to reduce their speed.
The branch of Physics that deals with phenomena such as those mentioned above is
called mechanics.
MECHANICS:
The branch of Physics that is concerned with the motion and equilibrium of bodies in
a particular frame of reference is called “mechanics”. Mechanics can be divided into
three branches: (i) Statics – which deals with bodies at rest relative to some given
frame of reference, with the forces between them and with the equilibrium of the
system; (ii) Kinematics - the description of the motion of bodies without reference to
mass or force; and (iii) Dynamics – which deals with forces that change or produce
the motions of bodies.
Some common terms used in the study of mechanics (and indeed many other branches
of Physics) are: scalars, vectors and SI Units.
1. Scalars – A scalar is a physical quantity defined in terms of magnitude (size)
only - eg temperature, mass, volume, density, distance.
2. Vectors – A vector is a physical quantity defined in terms of both magnitude
and direction - eg force, velocity, acceleration, electric field strength.
Diagramatically we can represent a vector by a straight line with an arrow on
one end. The length of the line represents the magnitude of the vector
quantity and the direction in which the arrow is pointing represents the
direction of the vector quantity. We will say much more about vectors later in
this topic.
3. Systeme International (SI) Units – The internationally agreed system of units.
There are seven fundamental units. The three that we will use in this topic are the
metre (length), the kilogram (mass) and the second (time). Various prefixes
are used to help express the size of quantities eg a nanometre (1 nm) = 10-9 of a
metre, a gigametre (1 Gm) = 109 metres. See Appendix C, p.243 of Excel
Preliminary Physics by Warren for a list of some common SI units and prefixes.
Most texts will contain such information.
Since describing the features of something is usually a lot simpler than
explaining how or why it works, we will start with a look at kinematics.
MOTION:
The following terms are commonly used to describe motion.
1. Displacement – is the distance of a body from a given point in a given direction.
It is a vector quantity. The SI unit of displacement is the metre (m)
2. Speed – The speed of a body is the rate at which it is covering distance. It is a
scalar quantity. The SI units are m/s, which can also be written as ms-1.
where vav = average speed, d = total distance travelled and t = total time taken to
travel distance d.
3. Velocity – The velocity of a body is its speed in a given direction. In other
words, velocity is the rate of change of displacement with time. It is a vector
quantity with the same SI units as speed.
where vav = average velocity, r = change in displacement and t = change in
time taken to achieve that change in displacement.
Another way to express average velocity is as the average of the initial and final
velocities.
where vav = average velocity, u = initial velocity of the body and v = final
velocity of the body. Note that this equation applies ONLY when the velocity
of the body is increasing or decreasing at a constant rate.
4. Acceleration – The acceleration of a body is the rate of change of the velocity of
the body with time. It is a vector quantity, with units of (metres/second)/second,
written as ms-2.
where aav = average acceleration, v = change in velocity of the body and t =
change in time over which the change in velocity took place. Alternatively, we
may write:
where aav = average acceleration, v = final velocity of the body, u = initial
velocity of the body, and t = time over which the change in velocity took place.
Note that a body accelerates when:
a. It speeds up;
b. It slows down;
c.
It changes direction.
MOTION GRAPHS:
Often the most effective way to describe the motion of a body is to graph it. Note
that in this section the variable “s” will be used to represent displacement instead of
“r”. You will find in Physics that these two variables are both in common use to
denote displacement.
Displacement-Time Graphs
These may be used to gain information about the displacement of an object at various
times or about the velocity of the object at various times.
Clearly, the gradient (slope) of a displacement-time graph gives the velocity.
Gradient =
S/t
= velocity
Note that a positive gradient implies a positive velocity and a negative gradient
implies a negative velocity.
For a curved displacement-time graph, the gradient of the tangent to the curve at a
particular point equals the gradient of the curve at that point, which in turn equals the
velocity of the object at that particular time. Such a velocity, that is, the velocity at a
particular instant in time, is called the instantaneous velocity.
An example of an instrument that measures instantaneous velocity is the speedometer
in a car. In older cars the speedometer was linked mechanically to the transmission.
These days, however, a device located in the transmission produces a series of
electrical pulses whose frequency varies in proportion to the vehicle's speed. The
electrical pulses are sent to a calibrated device that translates the pulses into the speed
of the car. This information is sent to a device that displays the vehicle's speed to the
driver in the form of a deflected speedometer needle or a digital readout.
Note that a straight line displacement-time graph implies that velocity is constant. A
curved line displacement-time graph implies that velocity is changing with time (ie
the object is accelerating).
Velocity-Time Graphs
These may be used to gain information about the displacement, velocity and
acceleration of an object at various times.
The gradient is clearly the acceleration of the object.
Gradient =
v/t
= acceleration
Note that a positive gradient implies a positive acceleration and a negative gradient
implies a negative acceleration.
Also, the area under the graph,
in the case above, has units of: seconds x metres per second = metres. Thus, the area
under a velocity-time graph is equal to the displacement travelled by the object
in the time t.
Note that a horizontal straight line velocity-time graph implies that acceleration is
zero – ie velocity remains constant.
A non-horizontal, straight line velocity-time graph implies that acceleration is
constant and non-zero.
A curved line velocity-time graph implies that acceleration is varying.
Acceleration-Time Graphs
These may be used to gain information about the velocity and acceleration of an
object at various times.
The area under an acceleration-time graph gives the velocity of an object. Check
the units of the area: (ms-2 x s = ms-1).
A horizontal straight line acceleration-time graph implies that velocity is varying at a
constant rate (ie velocity is increasing or decreasing by the same amount each
second). That is, acceleration is constant.
RELATIVE VELOCITY:
Often it is necessary to compare the velocity of one object to that of another. For
instance, two racing car drivers, A and B, may be travelling north at 150 km/h and
160 km/h respectively. We could say that the velocity of car B relative to car A is 10
km/h north. In other words, driver A would see driver B pull away from her with a
velocity of 10 km/h north.
Likewise, two jet aircraft, C and D, flying directly at each other in opposite directions
(hopefully as part of an aerobatics display) may have velocities of 900 km/h north and
1000 km/h south respectively. We could say that the velocity of D relative to C is
1900 km/h south. In other words, jet C will observe jet D flying towards it at a speed
of 1900 km/h.
Clearly, when the objects are travelling in the same direction, the velocity of one
relative to the other is the difference between their speeds, taking due care to state the
appropriate direction. When the objects are travelling in opposite directions the
velocity of one relative to the other is the sum of their speeds, again taking due care to
state the correct direction. There is a vector equation which can be used to calculate
the relative velocities of objects, even when the objects travel at various angles to one
another but this equation is outside the scope of the present syllabus (for some
unfathomable reason).
FORCE:
What is Force?
In simple terms a force can be defined as a push or a pull. We experience examples of
forces every day. If we push a stationary lawn mower (with enough force) it begins to
move – that is, it accelerates and its velocity increases. If we push on the brakes of a
moving bicycle, it slows down – that is it decelerates (or undergoes negative
acceleration). If we apply sufficient force to an aluminum can, by squeezing it with
our hand, we can change the shape of the can.
So a force can cause a change in the state of motion of an object or a change in the
shape of an object. In fact, all accelerations (and decelerations) are caused by forces .
Does every force cause acceleration?
Again, from our everyday experience, we know the answer to this question is “no”. If
a person pushes on the brick wall of a house, the house does not accelerate.
Sometimes when we want to push or pull an object from one place to another we find
that no matter how hard we push or pull, we just cannot move (accelerate) the object.
In previous Science courses, a qualitative relationship was established between force,
mass and acceleration. In the Preliminary Physics Course we need to establish a
quantitative relationship between these three quantities.
What is the relationship between force and
acceleration?
We could perform an experiment to determine the relationship between the size of a
force applied to an object at rest on a laboratory bench and the change in velocity
experienced by the object over a set period of time (ie the acceleration). Such an
experiment would produce results as shown below.
The graph above shows that:
 The change in velocity does not happen instantaneously. A certain amount of
force is required before the object begins to accelerate. This makes sense, since
the force of friction between the bench and the object must be overcome before the
object can move. So, we can say that a net external force is required in order to
change the velocity of an object.
 The acceleration produced is directly proportional to the force applied. If we
repeated the experiment on a frictionless surface (eg using a dry-ice puck on a
very smooth, polished table top) the straight-line graph would even pass through
the origin.
What is the relationship between acceleration and
mass?
We could measure the accelerations produced when the same sized force is applied to
different objects. Such an experiment would produce results like those below.
The graph above suggests that there is an inverse relationship between acceleration
and mass. A plot of acceleration versus the reciprocal of mass, using the same data,
would produce a graph similar to that below.
This graph clearly shows that:
 The acceleration produced by a given force is inversely proportional to the
mass of the object.
From experiments such as those above, we can say that:
and
NEWTON’S LAWS OF MOTION
Newton’s First and Second Laws:
By combining the results above and defining the units of force appropriately, we can
write that:
.
This can be taken as a statement of Newton’s Second Law. The SI Unit of force
is the newton (N), defined so that 1N = 1kgms-2.
Note that in the above equation, F is the vector sum of all the forces acting on the
object, m is the mass of the object and a is its vector acceleration. To remind us of
that fact we will write:
.
Note that if the resultant force on the object is zero, there is no acceleration.
Therefore, in the absence of a resultant force, an object’s velocity will remain
unchanged. In other words, an object at rest will remain at rest, and an object in
motion will remain in motion with uniform velocity, unless acted upon by a net
external force. This is a statement of Newton’s First Law, which in fact is
contained in the Second Law
Newton’s Third Law:
Forces acting on a body originate in other bodies that make up its environment. Any
single force is only one aspect of a mutual interaction between two bodies. We find
by experiment that when one body exerts a force on a second body, the second body
always exerts a force on the first. Furthermore, we find that these forces are equal in
size but opposite in direction. A single, isolated force is therefore an impossibility.
If one of the two forces involved in the interaction between two bodies is called an
action force, the other is called the reaction force. Either force may be called the
action and the other the reaction. Cause and effect is not implied here, but a mutual
simultaneous interaction is implied.
This property of forces was first stated by Newton in his Third Law: “To every
action there is always opposed an equal reaction; or, the mutual actions of two
bodies upon each other are always equal, and directed to contrary parts.”
In other words, if body A exerts a force on body B, body B exerts an equal but
oppositely directed force on body A; and furthermore the forces lie along the line
joining the bodies. Notice that the action and reaction forces, which always occur
in pairs, act on different bodies. If they were to act on the same body, we could
never have accelerated motion because the resultant force on every body would be
zero.
Consider the following examples:
1.
Imagine a boy kicking open a door. The force exerted by the boy B on the door
D accelerates the door (it flies open); at the same time, the door D exerts an equal
but opposite force on the boy, which decelerates the boy (his foot loses forward
velocity). The force of the boy on the door and the force of the door on the boy is
an action-reaction pair of forces.
2.
When you walk, you apply a force backwards on the earth. Likewise, the earth
applies a force to you of equal magnitude but in the opposite direction. So, you
move forwards.
The force of the person on the earth and the force of the earth on the person is an
action-reaction pair of forces.
3.
Consider a body at rest on a horizontal table:
Each of the pairs of forces above is an action-reaction pair of forces.
Definitions of Mass and Weight:
The mass of an object is a measure of the amount of matter contained in the
object. Mass is a scalar quantity.
The weight of an object is the force due to gravity acting on the object. Weight is
a vector quantity.
The weight, W, of an object is given by Newton’s 2nd Law as:
where m is the mass of the object and g is the acceleration due to gravity (9.8 ms-2
close to the earth’s surface).
USEFULNESS OF VECTOR DIAGRAMS:
Many of the quantities with which we deal in Physics are vectors. Sometimes we
need to add a number of vectors together. For instance, we may be trying to calculate
the total or resultant force acting on a car when several forces act on the car
simultaneously – the wind, friction, gravity and the force supplied by the engine.
Sometimes we need to subtract two vectors. For instance, we may be trying to
calculate the change in velocity of a car as it goes around a bend in the road. The
change in velocity of the car equals the final velocity of the car minus initial velocity
of the car.
When the need arises to add or subtract vector quantities, this proves to be easy
only when the vector quantities act along the same straight line. If the vectors
act at an angle to each other we really need to draw a vector diagram to assist in
solving the problem.
Vector analysis is an extremely important aspect of Physics and there are several
different methods available to add, subtract and even multiply vectors. The
current Syllabus requires that you have only a very basic understanding of
vector analysis. So, we will examine a single, simple but very useful method of
adding and subtracting vectors.
VECTOR ADDITION:
The method we will use is called the “Vector Polygon” method. To find the sum of
a number of vectors draw each vector in the sum, one at a time, in the appropriate
direction, placing the tail of the second vector so that it just touches the head of the
first. Continue in this fashion until all of the vectors in the sum have been included in
the diagram. Note that it does not matter which vector you start with.
The vector that closes the vector polygon in the same sense as the component vectors
is called the equilibrant. It is the vector which when drawn into the diagram gets you
back to where you started. The vector that closes the vector polygon in the
opposite sense to the component vectors is called the resultant. The resultant is
the answer to the sum of all the vectors. Its size can be calculated
mathematically or measured using a ruler if the vector polygon has been drawn
to scale. The direction of the resultant can be calculated mathematically or can
be measured using a protractor if the vector polygon has been drawn to scale.
Either way, the direction of the resultant must be stated in an unambiguous way.
Sometimes in Physics our vector additions only involve two vectors at a time. In this
case, the polygon formed is a triangle, making the mathematical calculation of the
magnitude and direction of the resultant quite straight forward. So, keep your wits
about you and bring your knowledge of triangle geometry and trigonometry into
the Physics classroom.
EXAMPLE 1: A fighter pilot flies her F-14D Tomcat jet with a true airspeed of 400
km/h North. A crosswind from the East blows at 300 km/h relative to the ground.
Calculate the jet’s resultant velocity relative to the ground.
Note: For aircraft, the true airspeed (TAS) is the actual speed of the aircraft
through the air (the speed of the aircraft relative to the air). The wind speed is
usually measured relative to the ground. Groundspeed is the speed of the aircraft
relative to the ground. The groundspeed of the aircraft is the vector sum of the
true airspeed and the wind speed.
Obviously, a vector diagram would be very useful in solving Example Problem 1.
See the diagram below.
By Pythagoras’ Theorem, the magnitude of the resultant velocity of the jet is:
and the direction can be found using basic trigonometry as follows:
So, the velocity of the jet relative to the ground is 500 km/h N36.9oW.
Note that if the angle between the two vectors being added together is other than 90o,
Cosine Rule and Sine Rule can be used to solve the problem mathematically. Note
also the use of the compass in the diagram to establish direction.
EXAMPLE 2: In the previous problem, in which direction should the pilot head
and with what airspeed in order to actually fly north at 400 km/h relative to the
ground?
Again a vector diagram is useful. Our intuition tells us that the pilot must fly into the
wind. So, when we draw a diagram that shows all of the information that we know to
be true, we obtain the diagram shown below.
Clearly, if the pilot flies N36.9oE with an airspeed of 500 km/h, the wind will
bring her back to a heading of due North at a speed of 400 km/h relative to the
ground. Remember also, there is usually more than one way to give the direction.
The direction the pilot should fly in this example could just as correctly be given as
E53.1oN or as a True Bearing of 36.9o.
EXAMPLE 3: Four children pull on a small tree stump firmly stuck in the ground.
Looking down on the tree stump from above, the forces applied by the children are as
shown below.
Determine the resultant force applied to the tree stump.
To solve this problem mathematically we would need to add two of the vectors
together, then add our answer to the third vector and finally add our answer to that
addition to the fourth vector. It is actually far quicker and easier to solve this problem
graphically. To do this we construct a vector polygon, using the rules stated above
and simply measure the size and direction of the resultant. See the diagram below.
The resultant force, R, is found by measurement to be 3.1 N at an angle
of 39o clockwise from the direction of the 4.5 N force.
Note that when using a graphical approach, the scale must be clearly
stated on the diagram. Always choose a sensible numerical scale. Also,
choose a scale that will produce a large diagram. The larger the diagram, the
more accurate the answer. For the example problem above, the scale used was 1
N = 1.5 cm. This is certainly the smallest scale I would use for this particular
problem. Anything less is too inaccurate. A scale of 1 N = 2 cm would be
preferable. The smaller scale was used here to fit the diagram neatly onto this
page.
VECTOR SUBTRACTION:
If vector A is as shown below:
then vector –A is a vector of the same magnitude as A but opposite direction.
In order to find the difference between two vectors, add the negative of the
second vector to the first.
EXAMPLE 4: A Maserati (car) is moving due East at 20 ms-1. A short time later it is
moving due North at 20 ms-1. Calculate the change in velocity of the Maserati.
To find the change in size of any quantity, you subtract the initial size of the quantity
from the final size. Obviously, with vector quantities you must do a vector
subtraction not just an arithmetic one, since vectors possess both size and direction.
Change in velocity = final velocity – initial velocity
This should really be written as:
Change in velocity = final velocity + (– initial velocity)
since that is how we draw the vector diagram. We literally add the negative of the
initial velocity to the final velocity. Study the vector diagram below to ensure you
understand the process of vector subtraction.
Using Pythagoras’ Theorem and basic trigonometry as shown in Example 1 above, we
find that the change in velocity of the Maserati is 28.3 ms-1 at an angle of 45o West
of North.
Note that even though the car has the same initial and final speeds, because the
direction of the car has changed, so too has its velocity.
VECTOR COMPONENTS:
Sometimes we are only interested in part of a vector rather than all of it. For instance,
if we push a car that has run out of petrol, we apply a force to the car. However, if we
are not careful some of the force we apply pushes down vertically on the car and the
rest of it pushes horizontally on the car. Obviously, we are trying to maximise the
component of the force that is applied horizontally. The angle at which we apply
force to the car will determine how much of our force is applied horizontally.
Any vector may be broken into two component vectors at right angles to each
other. These components are called the rectangular components of the vector.
The rectangular components of a vector add up to the original vector.
Consider, the example we used above. We may push on the car with a force F at an
angle of  to the horizontal, as shown below.
The force F may be broken into its vertical and horizontal components as shown
below.
The magnitude of each component can then be calculated using simple trigonometry.
The size of the
. The size of the horizontal
component of F, the one that must overcome the force of friction if we are to move
the car forward,
.
EXAMPLE 5: A block of mass 20 kg is being pulled up an inclined plane by a rope
inclined at 30o to the plane’s surface as shown in the diagram below.
The plane is inclined at 45o to the horizontal. The friction force, F, opposing the
block’s motion is 10 N. Determine the tension, T, in the rope if the net acceleration,
a, of the block up the plane is 4 ms-2.
You will observe that we have resolved two vectors in the diagram into rectangular
components – the tension, T, in the rope and the weight, W, of the block. The
rectangular components we have chosen are those acting parallel to and perpendicular
to the inclined plane. These components are the most useful ones in a situation like
this.
Now the total force acting down the plane is the sum of the friction force, F, and the
from the diagram we have:
FD = F + mg sin
FD = 10 + 20 x 9.8 x sin 45o
FD = 148.59 N
Total force acting up the plane must be:
FU = ma + FD
FU = (20 x 4) + 148.59
FU = 228.59 N
Note that the logic we have used to obtain an expression for FU is as follows. The
force up the plane MUST be sufficient to overcome the 148.59 N force down the
plane and to provide the required force of 80 N to give the block the correct
acceleration up the plane.
Now we are in a position to calculate the tension in the rope. The total force up
the plane FU is actually the component of the tension force parallel to the plane. This
is the part of the tension force that is applied parallel to the plane.
Therefore, we have:
cos 30o = FU / T
T = FU /cos 30o
T = 228.59 / cos 30o
T = 263.95 N
So, the tension in the rope works out to be 264 N.
Circular Motion
An object moving in a circular path at constant speed is said to be executing
uniform circular motion. Obviously, although the speed is constant, the velocity is
not, since the direction of the motion is always changing. It can be shown that for an
object executing uniform circular motion (UCM), the acceleration keeping the object
in its circular path is given by:
ac = v2/r
where ac is called the centripetal ("centre-seeking") acceleration, v = speed of the
object and r = radius of the circular path. As the name implies, centripetal
acceleration is directed towards the centre of the circle.
Using the fact that force can be written as F = ma, the centripetal force, Fc, acting on
an object undergoing UCM is given by:
where m = mass of the object. This force is also directed towards the centre of the
circle.
Example: A car of mass 900 kg moves at a constant speed of 25 m/s around a
circular curve of radius 50 m. Calculate the centripetal force acting on the car.
(11250 N, towards the centre of the circle)
WORK
A force applied to an object is often capable of moving that object through a certain
distance. Whenever this happens we say that work has been done on that object.
Work is a scalar quantity defined mathematically as:
where W = work done on object, F = force acting on object along the line of motion
and s = net displacement of object caused by the force along the line of motion.
The SI unit of work is the joule (J). 1J = 1 Nm
EXAMPLE: Calculate the work done when an object is moved through a
displacement of 20m north by a force of 10N north.
ANSWER: W = F.s and so W = 10 x 20 = 200J (Note that there is no direction, since
work is a scalar quantity.)
ENERGY
Energy and work are closely related quantities. An object can do work only if it has
energy. Energy, then, is the property of a system that is a measure of its capacity
for doing work. The amount of energy an object has is equal to the amount of
work it can do. Like work, energy is a scalar quantity with an SI Unit of the
joule (J).
Energy has several forms: electric energy, chemical energy, heat energy, nuclear
energy, radiant energy (ie EM radiation such as light), mechanical energy (eg kinetic
energy) and sound energy (ie the kinetic energy of the vibration of the air). In a
closed system (ie one in which no mass enters or leaves), energy can neither be
created nor destroyed, although it may be transformed from one form into
another. This is called the Law of Conservation of Energy.
KINETIC ENERGY
Kinetic energy is the name given to the energy associated with a moving body. It
can be shown that the amount of kinetic energy possessed by a body is given by:
where m = mass of the body and v = velocity the body.
Moving vehicles have kinetic energy. Consider a small car of mass 920 kg moving at
a constant speed of 60 km/h (16.67 m/s). The kinetic energy of the car can be
calculated as:
Ek = 0.5 x 920 x 16.672
Ek = 1.28 x 105 J
To change the velocity of a moving body, or to set a body at rest into motion, a
net force must be applied to it and work must be done on it. The work done on
the body is equivalent to the change in kinetic energy of the body.
POTENTIAL ENERGY
Stored energy is called potential energy, since it has the potential to do work for us.
Examples of potential energy include: the energy stored in a stretched (or
compressed) spring; the chemical energy stored in a car battery; the energy stored in
the water in a damn above a hydroelectric power station; and the energy stored in the
chemical bonds holding compounds together.
ENERGY TRANSFORMATIONS
Energy transformations (changes) are an important aspect in understanding motion.
In a car battery, chemical potential energy is transformed into electrical energy. In an
internal combustion engine (such as a car engine) the chemical potential energy stored
in petrol is transformed among other things into mechanical energy in the form of
kinetic energy of motion.
When cars collide, various energy transformations take place. Some of the kinetic
energy (KE) of the cars is transformed into sound – we hear the collision. Some KE
is changed into radiant energy (light energy) – we see sparks fly as the wreckage
scrapes along the ground. Some of the KE is transformed into heat – the friction
produced by metal scraping on metal and tyres gripping the road under heavy braking
produces heat. Some KE is transformed into potential energy of deformation – the car
bodies are compressed, compacted and twisted during the collision and some of the
KE is stored in the deformed wreckage. In a worst-case scenario, where an explosion
takes place, chemical energy stored in the fuel is converted into kinetic energy (and
sound, heat, light) as parts of the wreckage are flung far and wide.
MOMENTUM:
Everyday experience tells us that both the mass and velocity of an object are
important in determining things like (i) how hard it is to stop the object or (ii) the
effect the object has in a collision with another object. An 85 kg man running at 5
m/s is a lot harder to stop than a 15 kg six year old child running at the same speed. A
50 gram bullet fired from a rifle with a muzzle velocity of 500 m/s will do a lot more
damage than an identical bullet thrown at the target by hand.
Isaac Newton spoke of the “quantity of motion” of an object. Today we define the
momentum of an object to be the product of mass (m) and velocity (v).
Momentum is a vector quantity with SI Units of kgms-1 (or Ns, since 1N = 1kgms-2).
Newton’s 2nd Law can be re-written as:
change in momentum to occur.
This quantity
(the change in momentum) is given the name impulse. Clearly,
from the above equation, impulse, I, is defined as the product of force and time
and has SI Units of Ns. Impulse is a vector quantity.
CONSERVATION OF MOMENTUM
According to Newton’s 1st and 2nd Laws of motion, there is no change in momentum
without the action of a net external force. Thus, if no net external force acts on a
particular system, the total momentum of the system must be constant. This is
known as the principle of the conservation of linear momentum.
A system on which the net external force is zero is given a special name. Such a
system is called an isolated system. So, another way to express the principle of the
conservation of linear momentum is to say that within an isolated system the
total momentum is a constant. This principle is applicable to many important
physical situations.
CONSERVATION OF MOMENTUM DURING
COLLISIONS
One important physical situation to which the principle of the conservation of linear
momentum is applicable is the case of collisions between bodies. In such cases, if we
assume that no external net force acts during the collision, we can say that the total
momentum of the system before collision equals the total momentum of the
system after collision. This proves to be an extremely useful starting point for
analysing many collision situations.
To see that momentum is conserved during collisions we can use Newton’s 3rd
Law. Consider a collision between two particles, A and B, as shown below.
During the brief collision these particles exert large forces on one another. At any
instant FAB is the force exerted on A by B and FBA is the force exerted on B by A. By
Newton’s 3rd Law these forces at any instant are equal in magnitude but opposite
in direction.
The change in momentum of A resulting from the collision is:
in which the bar above the FAB indicates that we are taking the average value of FAB
The change in momentum of B resulting from the collision is:
in which the bar above the FBA indicates that we are taking the average value of FBA
Note that it is necessary to take the average value of the collision forces since the
magnitudes of both forces will vary over the duration of the collision.
If no other forces act on the particles, then
A and
B give the total change in
momentum for each particle. But we have seen that at each instant:
FAB = - FBA
So that:
And therefore that
:
pA = - pB .
If we consider the two particles as an isolated system, the total momentum of the
system is:
P = pA + pB
And the total change in momentum of the system as a result of the collision is zero,
that is:
P = pA + pB = 0.
Thus, using Newton’s 3rd Law and our knowledge of impulse we have shown that if
there are no external forces, the total momentum of the system is not changed by
the collision. Therefore, as we said before, if we assume that no external net force
acts during the collision, we can say that the total momentum of the system before
collision equals the total momentum of the system after collision.
How accurate is it though to assume that no external net force acts on a system during
a collision? When a golf club strikes a golf ball surely there are external forces that
act on the system of club + ball? Indeed there are: gravity and friction are two
obvious forces that act on both club and ball. So how can we simply ignore these
forces?
The answer is that it is safe to neglect these external forces during the collision
and to assume that momentum is conserved provided, as is almost always the
case, that the external forces are negligible compared to the impulsive forces of
collision. If the external forces are negligible compared to the impulsive forces, then
the change in momentum of a particle during a collision arising from an external force
is negligible compared to the change in momentum of that particle arising from the
impulsive force of collision.
In the case of the golf club striking the golf ball, the collision lasts only a tiny fraction
of a second. Since the observed change in momentum is large and the time of
collision is small, it follows from the impulse equation:
p = F t
that the average impulsive force F is relatively large. Compared to this force, the
external forces of gravity and friction are negligible. During the collision we can
safely ignore these external forces in determining the change in motion of the ball; the
shorter the collision time, the more accurate this assumption becomes.
In practice, we can apply the principle of momentum conservation during
collisions if the time of collision is small enough.
INERTIA
As we have seen, Newton’s 1st Law states that an object in uniform motion will
remain in uniform motion and an object at rest will remain at rest, unless acted upon
by an external, net force. This ability of a body to resist changes in its state of
motion is called the inertia of the body. The inertia of a car, for instance, is its
tendency to remain in uniform motion or remain at rest. The fact that bodies possess
inertia has important consequences when dealing with moving vehicles.
A moving vehicle is a complex body. It consists of the vehicle body itself, the driver
(and passengers) and other objects carried in the car. If the driver or passengers or
other objects are not restrained, they will continue to move at whatever speed the car
is travelling, even after the application of the brakes. Let us consider some questions:
What are some of the dangers presented by loose objects in vehicles?
Most people, when they see an unrestrained object fly off the car seat when they
slam on the brakes, would say that it got pushed off the seat (ie some sort of force
acted on the object to move it off the seat). Is this an accurate account of the
physics of the situation? Explain in terms of Newton’s 1st Law.
Why do you think Newton’s 1st Law of Motion is not applied correctly in many
real world situations (like the one mentioned above)?
What is the function of an inertia reel safety belt? Where could we obtain
information on how they work and their effectiveness?
Describe a possible experiment we could do to assess the relative effectiveness
of lap, lap-sash and harness seatbelts in reducing the effects of inertia in a
collision.
List the features of a modern car that are designed to reduce or avoid the effects
of a collision. Where would we obtain information to use in assessing the relative
effectiveness of these features?
Assess the reasons for the introduction of low speed zones in built-up areas and
the addition of air bags and crumple zones to vehicles with reference to the
concepts of impulse and momentum.