MATSEC AM Applied Mathematics cheat sheet

Constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t. These apply only when acceleration is uniform.

Variable acceleration: a = dv/dt = v·dv/ds; integrate rather than use suvat formulas. Displacement is the integral of velocity, and distance travelled requires splitting at turning points where v = 0.

Projectile motion: horizontal component u·cosθ is constant; vertical motion has acceleration −g. Time of flight T = 2u·sinθ/g on level ground; range R = u²·sin2θ/g; maximum range at θ = 45°.

Relative motion: velocity of A relative to B is v_A − v_B (vector subtraction). Closest approach occurs when the relative position is perpendicular to the relative velocity.

Newton's second law in vector form: ΣF = ma. Resolve along and perpendicular to the direction of motion; for a particle on an incline, use components mg·sinθ and mg·cosθ.

Connected particles: apply F = ma to each body separately and use the constraint that bodies joined by an inextensible string share the same magnitude of acceleration.

Motion on a rough incline: friction acts up the slope when the particle moves down. Net force down the slope = mg·sinθ − μmg·cosθ.

Momentum form of Newton's law: F = dp/dt = d(mv)/dt. Use this when mass changes, e.g. a leaking cart or a rocket.

Equilibrium of a particle: ΣF = 0, so the vector sum of resolved forces in any two perpendicular directions is zero.

Equilibrium of a rigid body: ΣF = 0 and ΣM = 0 about any point. Choose the point for taking moments to eliminate unknown forces through that point.

Moment of a force about a point = F × d, where d is the perpendicular distance from the point to the line of action. Units: N·m.

Couple: two equal, opposite, parallel forces separated by distance d produce a moment Fd independent of the pivot.

Limiting friction: F = μR, where R is the normal reaction. Static friction satisfies F ≤ μR; kinetic friction uses the coefficient of kinetic friction.

On an inclined plane, R = mg·cosθ and the component of weight along the plane is mg·sinθ. The angle of friction λ satisfies tanλ = μ.

Ladder problems: three unknowns (wall reaction, ground reaction, friction) require resolving horizontally and vertically plus taking moments about the foot or top.

A body topples when the line of action of its weight passes outside its base; sliding occurs first if μ is smaller than the ratio (base half-width)/(height of centre of mass).

Work done by a constant force = F·d·cosθ, where θ is the angle between force and displacement. Work–energy theorem: net work = change in kinetic energy.

Kinetic energy = ½mv²; gravitational potential energy change = mgh; elastic potential energy in a spring = ½kx².

Conservation of mechanical energy applies when only conservative forces do work; with friction, energy lost = work done against friction = μR·d.

Power = work/time = F·v for a constant force. A vehicle at maximum speed has driving force equal to total resistance.

Impulse = FΔt = change in momentum = mv − mu. Impulse is a vector and points in the direction of the change in momentum.

Conservation of linear momentum: total momentum before = total momentum after in the absence of external forces. Applies to all collisions and explosions.

Coefficient of restitution e = (relative speed of separation)/(relative speed of approach). e = 1 for perfectly elastic, e = 0 for perfectly inelastic collisions.

For a perfectly elastic collision, kinetic energy is conserved; for perfectly inelastic, the bodies coalesce and share a common velocity after impact.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.

Multiplication rule: P(A ∩ B) = P(A)·P(B|A). For independent events, P(B|A) = P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B). Bayes' theorem: P(A|B) = P(B|A)P(A)/ΣP(B|Aᵢ)P(Aᵢ).

Binomial distribution B(n, p): P(X = r) = C(n,r)·pʳ·(1−p)ⁿ⁻ʳ. Mean = np, variance = np(1−p). Assumes fixed n, independent trials, constant p.

Poisson distribution Po(λ): P(X = r) = e^(−λ)·λʳ/r!. Mean = variance = λ. Use as a limit of binomial when n is large and p small.

Normal distribution: standardise with z = (x − μ)/σ; use tables of Φ(z) = P(Z ≤ z). By symmetry, P(Z ≤ −z) = 1 − Φ(z).

Continuous random variable: P(a ≤ X ≤ b) = ∫f(x)dx from a to b; total probability ∫f(x)dx over the support equals 1.

Normal approximation to binomial: valid when np > 5 and n(1−p) > 5; use continuity correction, e.g. P(X ≥ k) ≈ P(Y ≥ k − 0.5).

Mean x̄ = Σx/n; for grouped data use class midpoints. Median is the middle value; mode is the most frequent value.

Variance s² = Σ(x − x̄)²/(n − 1) for a sample; population variance uses n. Standard deviation is the positive square root.

Coding: if y = (x − a)/b, then ȳ = (x̄ − a)/b and s_y = sₓ/|b|.

Quartiles: Q1, Q2, Q3. Interquartile range = Q3 − Q1. Outliers are typically values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Pearson's product-moment correlation coefficient r lies in [−1, 1]; r measures linear association only and is unaffected by coding.

Spearman's rank correlation coefficient rₛ = 1 − 6Σd²/(n(n² − 1)), used for ranked or non-linear monotonic data.

Least-squares regression line of y on x: y = a + bx, where b = S_xy/S_xx and a = ȳ − b·x̄. The line passes through (x̄, ȳ).

The regression line of y on x is not the same as x on y. Use y-on-x to predict y from x; the reverse requires the x-on-y line.

Sampling distribution of the mean: for a sample of size n from N(μ, σ²), x̄ ~ N(μ, σ²/n). The standard error is σ/√n.

Central limit theorem: for large n, x̄ is approximately normal regardless of the parent distribution.

Unbiased estimator of σ² from a sample: s² = Σ(x − x̄)²/(n − 1); use s/√n as the estimated standard error when σ is unknown.

Confidence interval for μ (σ known): x̄ ± z·σ/√n, where z = 1.96 for 95% and z = 2.576 for 99%.

Hypothesis test procedure: state H₀ and H₁, choose significance level, compute the test statistic, compare with the critical value, conclude in context.

One-sample z-test for μ: z = (x̄ − μ₀)/(σ/√n). Use a t-test with n − 1 degrees of freedom when σ is unknown and the sample is small.

Type I error: rejecting H₀ when it is true (probability = α). Type II error: failing to reject a false H₀.

Goodness-of-fit test: χ² = Σ(O − E)²/E with degrees of freedom = number of categories − 1 − number of estimated parameters.

Modelling cycle: formulate a real problem in mathematical terms, solve the model, interpret the solution, then validate and refine.

State all assumptions explicitly (e.g. particle treated as a point, air resistance neglected, constant acceleration). Assumptions determine the model's validity.

Use dimensional consistency to check equations: both sides of any physical equation must have the same base dimensions (M, L, T).

Scale and units: convert all quantities to SI before calculation (m, s, kg, N). Report answers with sensible significant figures and units.

Linear models take the form y = mx + c; exponential models y = Ae^(kx) apply to growth/decay. Linearise by plotting ln y against x to estimate k.

Interpret gradient and intercept in context: e.g. in a distance–time graph, gradient = speed; in a velocity–time graph, gradient = acceleration and area = displacement.

Evaluate a model by comparing predictions with observed data, identifying systematic errors, and stating limitations of the assumptions.

In multi-step problems, sketch a diagram, define variables, identify the relevant principle (equilibrium, energy, momentum, probability), and check reasonableness of the final answer.