MEI Conference, July 2008
Some misconceptions and errors commonly seen in M1
It is sometimes difficult to tell whether a student has gone wrong because of a slip or
because of a misconception. However, the misconceptions in the following list are
fairly widely held and when seeing them in a student’s work one should be suspicious
of there being more wrong than a simple slip.
Misconceptions
Vectors
Many students do not manipulate vectors with any confidence and do not have
a clear concept of a vector equation.
Use of column vectors instead of the ai + bj form can help.
Kinematics
Taking the direction of the motion of a particle from its position vector r
instead of its velocity vector v.
Not realizing that for a particle to be (instantaneously) at rest it is necessary
for v = 0.
Using constant acceleration formulae inappropriately and also using
distance = speed × time when the acceleration is not zero.
Confusion of x – y and t – y graphs so that the direction of motion of a particle
is described in terms of the shape of its t – y graph.
Forces
Newton’s third law is not well understood. Instances are
ƒ
ƒ
drawing false normal reactions (e.g. at the point of contact of a string with
a block),
wrong arguments in lift problems.
Dynamics
Wrongly applying Newton’s second law to connected particles. In particular,
not being able to find the force in a coupling and not being able to deal with
systems involving pulleys where a connecting string has sections in different
directions.
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Projectiles
Various wrong ideas often based on false assumptions of linearity. For
instance that half the maximum height is achieved in half the time taken to
reach the full height.
Things often not done well (but not necessarily because of a misconception)
General
The use of clear diagrams for questions on statics, dynamics or kinematics
(produced even if not instructed to draw them) so that positive directions (and
an origin) are properly defined, thus reducing the occurrence of sign errors etc.
Recognising all the forces present and when their sense is determined by the
situation (e.g. tension in strings, friction acting in a direction opposed to the
motion).
Recognising that the situation being modelled has changed (e.g. removal of
one force may change the value of another).
Correct resolution in sensible directions, especially the ones given in a
question.
Using vectors appropriately.
Interpretation
Students are expected to be able to explain a result using a concise and sound
mathematical argument with the correct use of mathematical language. They
are also expected to understand the modelling process and be able to relate
results obtained from the mathematical model to the situation being modelled.
Students frequently reveal an almost total lack of understanding of the
significance of their results. This is sometimes compounded by a lack of
command of technical language so that terms like force, work, energy etc are
often used as elegant variations in a piece of prose.
Students should be encouraged to use diagrams and equations to aid their
explanations.
Algebra, trigonometry and calculus
The algebraic demands of the mathematical models are not always negligible.
Some scenarios require models where at least one of the quantities involved is
a variable. Students whose simple algebra and trigonometry are not sound are
likely to produce a confused mess very early in such questions and are not
really ready for them.
Students should be able to deal efficiently and accurately with simultaneous
equations in two variables and with quadratic equations.
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Showing a stated result is true
Many examination questions ask the students to establish a given result in
order to ensure that they have the correct expression for further use. The
argument must be made very clear involving statements such as resolving
vertically, applying Newton's second law down the plane etc. Methods tend to
be especially sketchy when students are asked to formulate an equation of
motion.
Statics
Diagrams are often not clear enough to allow subsequent resolution, false
reaction forces are introduced and new diagrams are not produced when the
situation being modelled is changed.
Many students seem to be able to resolve forces in a given direction but not be
able to establish an equation given that there is equilibrium. Many students
cannot draw a polygon whose sides represent the forces acting in a given
situation; in particular, the triangle of forces is not often considered as a
method of solution of a three force problem.
Dynamics
Generally, Newton's second law is applied accurately to single particles or
bodies but connected particle problems reveal many misunderstandings.
Many students are unable to separate the component parts when there are
several particles present and then do not seem to be able to relate common
sense to their analysis and are content to have one part of a system in
equilibrium and a connected part accelerating.
Many students take friction to be opposing acceleration instead of motion. For
example, suppose a sledge is being pulled along horizontal ground by a string
and then the string breaks; the frictional force continues to oppose the forward
motion even though the sledge is decelerating.
Kinematics
Whether or not vector notion is being used, the constant acceleration formulae
are applied even when the acceleration is manifestly not constant. When
vector forms are required many students struggle with them.
When formulae are quoted there is frequent confusion with the current value
of 'u', 't ' etc, so in a question which divides the motion into two parts the
value taken for 'u' in the second part is likely to be that used in the first. The
absence of a defined origin and of the direction being taken as positive often
lead to confusion and sign errors.
Students need techniques such as the introduction of question specific
variables to help them avoid confusion with the notation.
Students should always draw diagrams to establish the origin and the positive
direction.
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Projectiles
Many problems seem to stem from students learning formulae for horizontal
range, maximum height or even the trajectory and then applying them
inappropriately. It is worth noting that examination questions are usually
structured to suggest an efficient method and no assumption is made that the
student has committed to memory, say, the general trajectory formulae.
Many students, when using the ‘uvast’ formulae, allow ‘u’, ‘t’, ‘v’ etc to stand
for different measures in the same question; this invariably leads to confusion.
As with general kinematics the students need techniques to help them avoid
confusion with the notation. I would use x for the horizontal position, x for
the general horizontal component of velocity, x0 for the initial horizontal
component of velocity and similarly for the vertical direction in terms of y.
Some teachers use u x , vx , s y to achieve the same distinction.
Many students do not recognise situations where the result v 2 = u 2 + 2as
applied to the vertical motion gives an immediate answer to the question.
The absence of a diagram indicating the direction taken as positive frequently
leads to sign errors, often followed by fudging. Many solutions have no
indication of method, often making it impossible to tell which part of the
motion is being considered.
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General
ƒ Apart from technical problems with the content of the units,
the work of many students suffers from
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Misconceptions and mistakes in M1
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ƒ poor arithmetic and working to too few figures
ƒ the inability to carry out simple algebraic manipulation efficiently and
accurately
ƒ poor notation
ƒ poor presentation
ƒ lack of diagrams or indication of an origin or the positive direction
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Accuracy and presentation
Common errors and misconceptions
ƒ There are three levels of accuracy and presentation that
are relevant
ƒ Particularly pernicious are the misconceptions that are
false simplifications that take the place of proper analysis
such as
ƒ enough for the student to work through a question
ƒ enough to communicate to others
ƒ enough to show
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ƒ thrust in tow-bar
ƒ trajectory arguments
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Some misconceptions: vectors
ƒ Students often make mistakes with vector equations
ƒ equating vectors to scalars
ƒ not giving the correct signs
e.g. A mass of 20 kg is subject to two forces, F N and 12 j N. Write
down its equation of motion.
Students think of the 12 j N as a resistance and, using Newton’s
second law write
F – 12 j = 20 a instead of F + 12 j = 20 a
This is particularly likely if the force is called a resistance
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Some misconceptions: vectors
ƒ Using column vectors helps to avoid loss of vector form
⎛ x ⎞ ⎛ 1 ⎞ ⎛ −2 ⎞
⎜ y ⎟ = ⎜ 2 ⎟ − ⎜ 5 ⎟ instead of (xi + yj) = (i + 2 j) − (−2i + 5 j)
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
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Some misconceptions: kinematics
Applications to statics
ƒ Information about the direction of vectors can lead to
confusion.
Example
Told that the forces F, G and H are in equilibrium, most
students will write down F + G + H = 0.
F
With the diagram,
G
most will write F + G = H.
ƒ Taking the direction of motion to be that of the position vector r
instead of the velocity vector v
ƒ Not using v = 0 as the condition for a particle to be
instantaneously at rest
ƒ Using constant acceleration and even zero acceleration formulae
inappropriately
ƒ Confusing x-y and t-y curves
H
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Mathematics in Education and Industry
Some misconceptions: forces
Some misconceptions: dynamics
ƒ Newton’s third law is not well understood
ƒ false normal reactions
ƒ wrong arguments in lift problems
ƒ Wrongly using N2L as F – mg = ma
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ƒ Wrongly applying N2L to connected particles
ƒ e.g. trucks in a train
ƒ e.g. connected particles not moving in the same line
ƒ Belief that, under deceleration, the force in a coupling is
always a thrust
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Mathematics in Education and Industry
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Some misconceptions: Projectiles
Some common errors: general
ƒ Wrongly believing that when a particle is fired at a fixed
speed from an origin and there are two trajectories passing
through a given point, that the particle
ƒ is rising on one trajectory and falling on the other,
ƒ was projected at an angle less than 45° in one case and more
than 45° in the other.
ƒ Poor diagrams for statics, dynamics and kinematics problems
ƒ leading to sign errors and inconsistency
ƒ Failure to recognise all the (relevant) forces present and
know when their sense is determined by the problem
ƒ Failure to recognise that the situation being modelled has
changed and so some forces already calculated have changed
values
ƒ Failure to resolve correctly in sensible directions; in given
directions (and in clever directions)
ƒ Using vectors inappropriately
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Mathematics in Education and Industry
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Some common errors: kinematics
Some common errors: statics
ƒ
ƒ
ƒ
ƒ
ƒ Poor diagrams are at the root of many of the problems
Inappropriate use of constant (or zero) acceleration formulae
Loss of vector forms
Sign problems because of poor diagrams or no diagrams
Confusion of the meaning of s, u, v, a and t when used
repeatedly in a problem
ƒ encourage students to use working variables
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ƒ There are forces missing
ƒ There are ‘phantom’ reaction forces
ƒ Fresh diagrams are not drawn when the model is changed
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Some common errors: statics
Some common errors: dynamics
ƒ Labelling is often poor
ƒ forces with the same magnitude should have the same label
ƒ forces that do not necessarily have the same magnitude must
have different labels
ƒ Problems arising from missing forces and poor diagrams
ƒ Poor techniques for dealing with connected particles
ƒ Taking the direction of the friction force to oppose the
acceleration instead of the direction of motion
Example
Suppose a sledge is being pulled along horizontal ground by a
string and then the string breaks; the frictional force
continues to oppose the forward motion even though the
sledge is decelerating.
ƒ Some students resolve but do not write down equations for
the equilibrium
ƒ Many students are not familiar with the triangle of forces
method
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Some common errors: projectiles
Some common errors: algebra, trigonometry, calculus
ƒ Inappropriate use of standard formulae
ƒ Confusion of the meaning of u, v, s, a and t is the horizontal
and vertical directions and in different parts of the solution
ƒ Not recognising when it is not best to work via t as a
parameter
ƒ Absence of diagram leading to sign errors
ƒ Lack of indicated method often making it impossible to work
out what part of the motion is being considered
ƒ Some scenarios will generate models where one or more
quantities are variable, leading to linear, quadratic and
simultaneous linear equations.
ƒ Many students produce little more than alphabetical soup
thickened with a little fudge
ƒ Students should be able to deal efficiently with quadratic and
simultaneous linear equations.
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Some common errors: showing results
Some common errors: interpretation
ƒ Examination questions frequently ask the students to
establish a given result in order to ensure that they have the
correct expression for further use.
ƒ The argument must be made very clear involving statements
such as
ƒ resolving vertically,
ƒ applying Newton's second law down the plane etc.
ƒ Students should be able to explain a result using a concise
and sound mathematical argument with the correct use of
mathematical language.
ƒ Students are expected to understand the modelling process
and be able to relate results obtained from a model to the
situation being modelled.
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Some common errors: interpretation
ƒ Students are typically poor at recognising the significance of
what they have done and rarely form sound arguments or
use mathematical language well.
ƒ Students should be encouraged to use diagrams and
equations to aid their explanations
It is easy to show that if a body is projected along a rough surface, the
distance it slides before coming to rest depends on (the square of) its initial
speed but not on its mass; when asked about the significance of the
absence of the mass term, students tend to say things like it cancelled out
but do not make the point that the distance it slides is independent of the
mass.
Drawing and labelling diagrams in mechanics
Clear, accurate diagrams are an essential step in the correct analysis of a variety of mechanics
problems. They provide a means of recording the information given, make it possible to define an
origin and a positive direction and frequently reveal aspects of the situation that suggest efficient
methods of solution. Solutions attempted without diagrams frequently lead to inconsistencies in the
analysis. Poor or wrong diagrams can easily lead to false assumptions being made.
Force diagrams
1
Decide on a single particle or a body model according to whether
any moments are involved.
N
T
If no moments are to be considered a single particle model is
always adequate; this may involve representing
W
one body by one particle (e.g. a car),
several bodies each by one particle (e.g. a car and caravan connected by a tow-bar).
The single particle may be placed inside a representation of the object if this aids clarity.
You may always use this model in MEI M1.
T
If moments are involved, use a body model and consider the lines of
action of the forces. Note that R has not been marked as passing
through the centre of mass.
R
F
It often helps clarity slightly to separate items, e.g. the block and the
slope.
W
This model may be necessary in MEI Mn, n >1, but often a single
particle model is adequate.
2
Follow any instructions given in a question. You may be asked to mark in all the forces or only
those with components in a particular direction. If in doubt mark in all the forces acting.
The following are subject to point 2.
3
The diagram should represent all the forces acting at a point, or on a body or system.
Identify that point, body or system and do not include any forces not acting on it.
Example
A particle of mass m is attached to one end of a light string. The other end of the string is
fixed to a ceiling. The particle is in equilibrium. Mark in all the forces acting on the particle.
T
The forces acting on the particle
hanging from a ceiling are shown
on the left. The extra force shown
on the right does not act on the
particle and should not be
included.
T
mg
mg
Diagrams in mechanics
T
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It is often better not to indicate an internal force pair on a single diagram. If you have to consider
separately the forces acting on two or more bodies connected or in contact, it is often clearer to
use linked diagrams instead of a single diagram. Clearly label the pairs of forces given by N3L
with equal magnitudes and opposite directions.
4
Put in any forces given in the question, arrowed in the direction of the force and with any given
magnitudes marked. Indicate any angles given.
5
In dynamics problems indicate the direction(s) you are taking to be positive.
Example
A car of mass 950 kg and with a driving force of 500 N is pulling a trailer of mass
500 kg. The coupling between the car and the trailer is light, horizontal and rigid. The resistances
to the motion of the car and the trailer are 100 N and 50 N respectively. Draw a diagram showing
the forces acting on the car and trailer in the direction of motion, including the internal force in
the coupling.
acceleration a
100 N
50 N
trailer 500 kg
car 950 kg
TN
500 N
TN
T N is the internal force in the coupling written as a tension
Note that this is effectively a two particle model using linked diagrams
6
If any of the particles or bodies has a mass, mark and label the corresponding weight. In a body
model the weight must act through the centre of mass.
7
If any surfaces are in contact,
(a)
mark in the normal reaction at 90° to the tangent of contact,
(b)
determine whether or not the surfaces are smooth; if not, indicate a friction
force along the tangent of contact in the direction opposite to the movement of the
body (or opposite to the direction the movement would take place if there were no
friction).
NB1
The friction force opposes movement (i.e. velocity) not acceleration.
NB2
If the direction of the friction force cannot be determined produce diagrams for both cases.
direction of motion
Example A block of mass m is pulled
along a horizontal plane by a string at 30°
to the horizontal.
R
T
30°
F
mg
Diagrams in mechanics
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8
If any strings are involved, mark in the force in the line of the string as a tension.
9
If there are any hinges and resolution is required, it is usually better to mark in the reaction force
as two components at right angles instead of a single force at an unknown angle. However, the
reaction at the hinge is better left as a single force if this reduces the problem to one of only three
forces, for which special techniques may be employed.
General points to check.
10
Ensure all forces have labels and arrows; if the direction of a force is known, mark it as being in
that direction. Do use the same label for any forces with known equal magnitudes. Do not use the
same label for forces not known to have the same magnitude. Apply this rule to linked diagrams
as well as single diagrams.
11
Make sure that you have not made any unwarranted assumptions about magnitudes and directions.
In particular
Never mark in friction as taking its limiting value; make this statement in your working only after
checking that it is true. In general F ≤ Fmax = μR .
When dealing with a normal reaction in a body model, consider carefully its line of action; it may
well not pass through the centre of mass.
12
For a statics problem, having completed your force diagram, consider whether a force
polygon will help your analysis.
For a three force statics problem, remember that the forces must be concurrent and consideration
of this fact as well as the triangle of forces will often provide a simple method of solution.
13
We often want to show forces, velocities and accelerations on the same diagram. The best way to
do this without leading to confusion is to use different sorts of arrows. There is no standard
convention but I have used
single arrows for forces
double arrows for acceleration
blocked arrows for velocity
Diagrams in mechanics
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Diagrams for kinematics problems
These are often omitted and much confusion follows. It is always necessary to establish the
direction being taken as positive and often necessary to note the position of the origin. A particularly
important application is to projectile motion where the sense of positive and the position of the
origin may well not be clear from the context.
Example
A small stone is projected downwards at 30° to the horizontal from the top of a vertical
cliff that is 80 m above plane horizontal ground. Its initial speed is 20 m s −1 .
Note that with the convention of the diagram, the initial value
of y is 80, the initial vertical component of velocity is negative,
g is in the negative direction and that the stone hits the sea
when y = 0.
30°
20 m s −1
80 m
y
g
Suppose that the origin were taken at the top of the cliff, still
with y upwards. Now the initial value of y is zero and the
stone hits the sea when y = −80.
Suppose now that the positive value of y is downwards with the
origin at the top of the cliff. The initial value of y is zero, the
initial vertical component of velocity is positive, g is in the
positive direction and the stone hits the sea when y = 80.
O
x
Suppose that the positive value of y is downwards with the origin
at the bottom of the cliff.........
A clear diagram defining the origin and positive directions is
vital!
Diagrams for impulse, momentum and impact problems
We are dealing with vector quantities and so once again it is vital that the sense of each such quantity
clear. It is also vital that the direction taken to be positive should be clearly marked.
Example
Two equal spheres A and B, each of mass m are on a smooth horizontal table. Initially,
sphere A is at rest and sphere B is travelling directly towards it with a speed u. The coefficient of
restitution in the subsequent collision is e. Find the impulse acting on A in the collision.
before
u
0
B
A
m
m
+ ve sense
after
vB
Diagrams in mechanics
vA
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