MATSEC IM Applied Mathematics cheat sheet

Kinematics with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t. Use the equation omitting the quantity not given and not asked for.

Newton's second law in vector form: resultant force F = ma. Resolve along and perpendicular to the direction of motion; on an incline, components of weight are mg sin θ (down slope) and mg cos θ (into slope).

Friction model: limiting friction F = μR where R is the normal reaction; at rest the actual friction force takes whatever value (≤ μR) is needed to maintain equilibrium.

Projectiles: horizontal and vertical motions are independent. With no air resistance, horizontal velocity is constant and vertical acceleration is g downward; time of flight follows from vertical motion alone.

Momentum is conserved in all collisions and explosions: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Coefficient of restitution e = (relative speed of separation)/(relative speed of approach), 0 ≤ e ≤ 1.

Work, energy and power: work = Fd cos θ, kinetic energy = ½mv², gravitational potential energy change = mgh, power = work done per unit time = Fv when force and velocity are parallel.

Moments: for a body in equilibrium under coplanar forces, the sum of moments about any point is zero and the vector sum of forces is zero. Take moments about a point that eliminates an unknown force.

Circular motion: angular speed ω = v/r, centripetal acceleration a = v²/r = rω² directed toward the centre; the centripetal force is provided by the resultant of real forces (tension, gravity, friction), never an extra force.

Differentiate displacement to get velocity and velocity to get acceleration; integrate acceleration to recover velocity and displacement, using initial conditions to fix constants of integration.

Relative velocity: velocity of A relative to B is v_A − v_B (vector subtraction). Closest approach occurs when the relative velocity is perpendicular to the line joining the two bodies.

Equilibrium of three non-parallel coplanar forces requires them to be concurrent; a triangle of forces can then be solved with the sine rule or cosine rule.

For a body on the point of toppling, the normal reaction acts through the pivot edge and the moment of the weight about that edge just balances the overturning moment.

Impulse equals change in momentum: J = ∫F dt = mv − mu; on a force–time graph the impulse is the area under the curve.

Elastic strings and springs: tension T = λx/l where λ is the modulus of elasticity, x the extension and l the natural length; elastic potential energy = λx²/(2l).

Simple harmonic motion: acceleration is proportional to displacement and directed toward the fixed point, a = −ω²x; period T = 2π/ω, independent of amplitude.

Centre of mass of a uniform lamina or composite body is found by summing (area × centroid distance) and dividing by total area; a body suspended freely hangs with its centre of mass vertically below the point of suspension.

In a differential-equation model, separate variables or use an integrating factor; the arbitrary constant is fixed by an initial condition, and the solution should be checked for physical plausibility.

Always state units and significant figures consistent with the data given; answers to a mechanics problem should be checked for direction as well as magnitude.