MATSEC IM Applied Mathematics common mistakes
MATSEC IM Applied Mathematics · Understand the traps and how to avoid them.
When resolving a force, using sin for the component adjacent to the stated angle and cos for the opposite one, or mixing the two between separate parts of the same diagram.
Candidates memorise 'horizontal = cos, vertical = sin' from horizontal-inclination examples and then apply it mechanically when the angle is marked against the vertical or against the plane, instead of re-deriving each component from the right-angled triangle in front of them.
What to do instead
Before resolving, draw the right-angled triangle for each force and label the side adjacent to the given angle; write the component along that side as F cos θ and the perpendicular one as F sin θ, then check the limiting cases θ = 0° and θ = 90° to confirm the formula gives the physically sensible answer.
Writing down the equilibrium or motion equations without a labelled force diagram, so that a reaction, friction, or tension is omitted or drawn in the wrong direction, and the resulting equations are internally inconsistent.
Under time pressure candidates jump straight to algebra, treating the diagram as optional decoration rather than the source of the equations; sign errors and missing forces then become invisible because there is nothing to check the algebra against.
What to do instead
Make a free-body diagram compulsory for every statics or dynamics question: isolate each body, mark every contact force, weight, and applied force with an arrow and a symbol, choose and state positive directions, and only then write ΣF = ma or ΣF = 0 component by component, checking each symbol in the diagram appears in exactly one equation per direction.
Applying v = u + at, s = ut + ½at², and v² = u² + 2as to a situation where acceleration is not constant, for example a body whose driving force or resistance varies with speed, and quoting a single numerical answer.
The constant-acceleration equations are the most heavily drilled formulae in the course, so they are reached for automatically whenever motion is mentioned; the condition 'a is constant' is treated as a formality rather than a premise that must be verified.
What to do instead
Before selecting a suvat equation, explicitly state the acceleration and justify that it is constant (constant resultant force). If the force depends on v or t, set up the differential equation from Newton's second law, separate variables, and integrate with the initial conditions, or use impulse–momentum and work–energy instead.
In projectiles and relative-motion problems, treating the horizontal and vertical motions as coupled, or adding speeds as scalars when finding a relative velocity or a resultant velocity at an angle.
Everyday experience suggests a faster horizontal throw 'keeps it up longer', and vector addition is learned as a formula rather than as a triangle or parallelogram, so magnitudes get added directly and directions are lost.
What to do instead
Resolve the initial velocity once into u cos θ and u sin θ, then treat the two directions as completely independent: constant velocity horizontally, constant acceleration vertically, with the same time t in both. For relative velocity draw the vector triangle and use the cosine rule or components; never add magnitudes unless the vectors are parallel.
Using F = μR with R set equal to the weight regardless of other vertical forces, and assuming friction always equals μR rather than recognising it as a limiting value.
Textbook examples on horizontal ground make R = mg look like a definition of R, and the phrase 'coefficient of friction' is misread as a fixed ratio that applies in all states of motion, including equilibrium where friction is below its maximum.
What to do instead
Always obtain R from the equilibrium equation perpendicular to the surface, including any component of an applied force or of the weight along that perpendicular. Then distinguish the two regimes: in limiting equilibrium or sliding, F = μR; in static equilibrium short of slipping, determine F from the along-surface equilibrium equation and check only that F ≤ μR.
Treating a pulley system or two connected particles as a single body with one acceleration equation, or giving both particles the same acceleration sign without regard to the direction of motion.
The string constraint is understood qualitatively ('they move together') but not translated into a kinematic relation, and candidates dislike writing two separate equations of motion, so they collapse the system into one and lose the internal tension.
What to do instead
Write a separate equation of motion for each particle using its own free-body diagram, relate the accelerations through the inextensible string (equal magnitudes, directions linked along the string), then solve the simultaneous equations for a and T. Check the answer by substituting back into both equations and by confirming T > 0.
In momentum questions, ignoring the vector nature of momentum and the direction of impulse, or applying conservation of momentum to a collision where an external impulse (friction, a fixed wall) acts during the impact.
Momentum is often practised only in one-dimensional head-on cases with all motion to the right, so sign conventions are never forced; the condition for conservation is remembered as a label rather than as 'no net external impulse'.
What to do instead
Draw before-and-after diagrams, define a positive direction explicitly, and write every velocity with its sign. State why momentum is conserved (no external impulse along the line of impact) before using it; if a wall or rough surface acts, use impulse–momentum for the body instead. For two-dimensional impacts, conserve momentum separately along and perpendicular to the line of impact.
Differentiating or integrating a kinematic or force expression without regard to the chain rule and the variable of differentiation, for example writing a = dv/dt as a constant when v depends on x, or dropping the constant of integration.
Calculus is applied as a symbol-manipulation routine transplanted from pure mathematics, and the physical meaning of each derivative (rate with respect to time) is not connected to the composite dependence of the quantity on time.
What to do instead
Write the defining relation explicitly, a = dv/dt = (dv/dx)(dx/dt) = v dv/dx, and choose the form that matches the given information (v as a function of t or of x). Carry the arbitrary constant through and determine it from the stated initial conditions, then check the result by differentiating back and by testing its physical plausibility.
In work, energy, and power problems, omitting a term from the energy budget (work done against friction, gain in potential energy, the kinetic energy of a second connected body) or equating power to force times velocity when the force is not along the velocity.
Energy methods are attractive precisely because they avoid resolving forces, which tempts candidates to skip the inventory of stores and transfers; power is memorised as P = Fv without the condition that F is the component along the direction of motion.
What to do instead
Write an explicit energy statement: initial KE + initial PE + work done by applied forces = final KE + final PE + work done against friction, listing every term with its sign and including every moving body. For power use P = F·v, taking the component of force along the velocity, and for average power use total work divided by total time.
Presenting a final answer with no units, with inconsistent units (mixing grams and kilograms, centimetres and metres, or degrees and radians), or rounded so severely that the significant figures misrepresent the precision of the data.
The arithmetic is done in whatever units the numbers happened to be given in, and unit consistency and significant figures are treated as cosmetic rather than as part of the physics; intermediate rounding compounds the error.
What to do instead
Convert all data to SI at the start and carry units through each line of working as a built-in check. Keep full precision in the calculator, round only the final answer to the appropriate number of significant figures (usually three, or matching the least precise datum), and state the unit and direction where the quantity is a vector.