MATSEC AM Applied Mathematics flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Constant acceleration equations

For uniform acceleration a: v = u + at, s = ut + ½at², v² = u² + 2as and s = ½(u + v)t. Valid only when acceleration is constant, not for variable acceleration problems.

Variable acceleration relation

Acceleration is the derivative of velocity: a = dv/dt = v·dv/ds. Integrate to recover velocity or displacement; split integrals at points where v = 0 to find total distance.

Projectile range formula

For projection speed u at angle θ on level ground, range R = u²·sin2θ/g and time of flight T = 2u·sinθ/g. Range is maximised at θ = 45° when air resistance is neglected.

Relative velocity

Velocity of A relative to B is v_A − v_B. For closest approach, the relative position vector is perpendicular to the relative velocity vector; solve by setting their dot product to zero.

Newton's second law

The resultant force on a particle equals mass times acceleration: ΣF = ma. Resolve forces along convenient perpendicular directions, often parallel and perpendicular to motion.

Moment of a force

The turning effect about a point equals force times the perpendicular distance from the point to the line of action: M = Fd. Units are newton-metres (N·m).

Rigid body equilibrium

A rigid body is in equilibrium when the vector sum of forces is zero and the sum of moments about any point is zero. Taking moments about a point eliminates forces through it.

Limiting friction law

When a body is on the point of sliding, friction equals μR, where μ is the coefficient of friction and R is the normal reaction. Below limiting, friction satisfies F ≤ μR.

Angle of friction

The angle of friction λ satisfies tanλ = μ. On an incline at angle θ, sliding begins when θ exceeds λ, since the driving component mg·sinθ then exceeds limiting friction μmg·cosθ.

Couple moment

Two equal, opposite, parallel forces separated by perpendicular distance d form a couple with moment Fd. The moment is the same about every point in the plane.

Work–energy theorem

The net work done on a particle equals its change in kinetic energy: W_net = ½mv² − ½mu². Include work done by gravity, friction and applied forces.

Power formula

Power is the rate of doing work: P = W/t, or P = Fv for a constant force moving at speed v. At maximum speed, driving force equals total resistive force.

Impulse–momentum theorem

Impulse equals change in momentum: FΔt = mv − mu. Impulse is a vector, and its direction is that of the change in momentum, not necessarily the initial velocity.

Conservation of momentum

In the absence of external forces, total linear momentum is conserved in any collision or explosion. This holds whether or not kinetic energy is conserved.

Coefficient of restitution

For two colliding bodies, e = (relative speed of separation)/(relative speed of approach). e = 1 for perfectly elastic collisions; e = 0 when bodies coalesce.

Conditional probability formula

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. Rearranging gives the multiplication rule P(A ∩ B) = P(A|B)·P(B) for combined events.

Binomial distribution conditions

B(n, p) requires a fixed number n of independent trials, two outcomes per trial, and constant success probability p. Mean is np and variance is np(1 − p).

Poisson distribution

Po(λ) models counts of rare independent events in a fixed interval: P(X = r) = e^(−λ)λʳ/r!. Mean and variance both equal λ; it approximates B(n, p) when n is large and p small.

Standardising a normal variable

If X ~ N(μ, σ²), then Z = (X − μ)/σ ~ N(0, 1). Use standard normal tables for Φ(z) = P(Z ≤ z); by symmetry, P(Z ≤ −z) = 1 − Φ(z).

Sample variance divisor

The unbiased sample variance is s² = Σ(x − x̄)²/(n − 1), dividing by n − 1 rather than n. Population variance uses n. Standard deviation is the positive square root.

Coding effect on statistics

Under coding y = (x − a)/b, the mean transforms as ȳ = (x̄ − a)/b and the standard deviation as s_y = sₓ/|b|. Correlation r is unchanged by linear coding.

Pearson correlation coefficient

Pearson's r measures the strength of linear association between two variables, taking values from −1 to 1. It is unaffected by coding but sensitive to outliers.

Least-squares regression line

The line of y on x is y = a + bx with b = S_xy/S_xx and a = ȳ − b·x̄. It always passes through the mean point (x̄, ȳ).

Spearman rank correlation

Spearman's coefficient rₛ = 1 − 6Σd²/(n(n² − 1)) uses ranks rather than raw values. It measures monotonic association and is less affected by outliers than Pearson's r.

Standard error of mean

For a sample of size n from a population with standard deviation σ, the standard error of x̄ is σ/√n. Larger samples give more precise estimates of the population mean.

Confidence interval for mean

When σ is known, a 95% confidence interval for μ is x̄ ± 1.96σ/√n; for 99%, use 2.576. Use the t-distribution when σ is unknown and n is small.

Type I error

A Type I error occurs when a true null hypothesis is rejected. Its probability equals the significance level α chosen for the test, commonly 0.05 or 0.01.

Type II error

A Type II error occurs when a false null hypothesis is not rejected. Its probability is denoted β; power of a test is 1 − β, the chance of correctly rejecting a false H₀.

Chi-squared goodness-of-fit

The statistic χ² = Σ(O − E)²/E compares observed and expected frequencies. Degrees of freedom equal categories minus one, minus the number of parameters estimated from the data.

Modelling assumptions

Explicit assumptions such as treating a body as a particle, neglecting air resistance or assuming constant acceleration simplify the model. They must be stated and their validity discussed.

Dimensional consistency

Both sides of a physical equation must have identical base dimensions in mass, length and time. Checking dimensions detects errors and can suggest the form of a relationship.

Linearising an exponential model

For y = Ae^(kx), plot ln y against x. The graph is a straight line with gradient k and intercept ln A, allowing parameters to be estimated from experimental data.

Gradient of a velocity–time graph

On a velocity–time graph, the gradient equals acceleration and the area under the curve equals displacement. On a distance–time graph, the gradient equals speed.