[v] = L T^-1, [u] = L T^-1, and [at] = (L T^-2)(T) = L T^-1. Every term has the same dimension L T^-1, so the equation is dimensionally consistent (homogeneous).

[a] = L T^-2, [t] = T

Impulse is the product of a force and the time for which it acts: J = F * (delta t). By the impulse-momentum theorem it equals the change in momentum: J = delta p = m v - m u. SI unit: N s (= kg m/s).

J = F * delta t = delta p

The work-energy theorem states that the work done by the net force on a body equals the change in its kinetic energy: W_net = (1/2) m v^2 - (1/2) m u^2. If net work is positive the body speeds up; if negative, it slows down.

W_net = (1/2)m v^2 - (1/2)m u^2

Moment of inertia is the rotational analogue of mass: I = sum of (m_i r_i^2). It measures a body's opposition to a change in its rotational motion. SI unit: kg m^2. It depends on (i) the mass and its distribution and (ii) the position/orientation of the axis of rotation.

I = sum(m_i r_i^2)

Since g = G M / (R + h)^2, increasing the height h increases the distance from the centre of the Earth, so g decreases. For small heights, g' = g (1 - 2h/R) approximately.

g' = g (1 - 2h/R)

Young's modulus is the ratio of longitudinal (tensile) stress to longitudinal strain, within the elastic limit: Y = (F/A) / (delta L / L). SI unit: N/m^2 (pascal, Pa).

Y = (F L) / (A * delta L)

For the streamline (steady, non-viscous, incompressible) flow of a fluid, the sum of the pressure energy, kinetic energy and potential energy per unit volume is constant: P + (1/2) rho v^2 + rho g h = constant. Thus where the speed is high, the pressure is low.

P + (1/2) rho v^2 + rho g h = constant

The first law is the law of conservation of energy for a thermodynamic system: delta Q = delta U + delta W. Here delta Q is the heat supplied to the system (positive if absorbed), delta U is the increase in internal energy, and delta W is the work done by the system (positive if the gas expands).

delta Q = delta U + delta W

In a transverse wave the particles of the medium vibrate perpendicular to the direction of wave propagation (example: a wave on a stretched string, light waves). In a longitudinal wave the particles vibrate parallel to the direction of propagation, forming compressions and rarefactions (example: sound waves in air).

v = f * lambda

Newton's second law: the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. F = dp/dt = d(mv)/dt. For constant mass, F = m (dv/dt) = m a.

F = dp/dt = m a

Systematic errors have a definite cause and always shift readings in one direction (e.g. a zero error in an instrument, or a mis-calibrated scale); they can be minimised by correcting the instrument. Random errors occur due to unpredictable fluctuations and vary in size and sign; they are reduced by taking many readings and averaging.

Distance is the total length of the actual path travelled; it is a scalar and is always positive. Displacement is the shortest straight-line vector from the initial to the final position; it is a vector and can be positive, negative or zero. Displacement magnitude is always less than or equal to the distance.

In uniform circular motion, centripetal acceleration is the acceleration directed towards the centre of the circle that continuously changes the direction of velocity. Its magnitude is a_c = v^2 / r = omega^2 r, and it always points radially inward (towards the centre).

a_c = v^2 / r = omega^2 r

Newton's third law: to every action there is an equal and opposite reaction, and the two forces act on different bodies. Example: when we walk, the foot pushes the ground backward (action) and the ground pushes the foot forward (reaction), which moves us ahead.

F(AB) = - F(BA)

Static friction acts on a body at rest and is self-adjusting up to a maximum value (limiting friction) f_s(max) = mu_s N. Kinetic (sliding) friction acts on a moving body and has a nearly constant value f_k = mu_k N. For the same surfaces, mu_k < mu_s, so it is harder to start motion than to keep it going.

f_s(max) = mu_s N ; f_k = mu_k N

Work done by a constant force is the dot product of force and displacement: W = F s cos(theta), where theta is the angle between them. (a) W = 0 when theta = 90 degrees (force perpendicular to displacement, e.g. centripetal force). (b) W is negative when theta is obtuse (90 to 180 degrees), e.g. friction opposing motion.

W = F s cos(theta)

A conservative force does work that depends only on the initial and final positions, not on the path, and the work done in a closed loop is zero (example: gravity, spring force). A non-conservative force does path-dependent work and dissipates energy (example: friction, viscous drag).

In an elastic collision both linear momentum and kinetic energy are conserved (e.g. collisions between hard steel balls, or gas molecules). In an inelastic collision momentum is conserved but kinetic energy is not - some is lost as heat, sound or deformation; in a perfectly inelastic collision the bodies stick together.

Torque is the turning effect of a force about an axis, equal to the product of the force and the perpendicular distance of its line of action from the axis: tau = r F sin(theta) = r x F (cross product). SI unit: newton metre (N m). It is a vector along the axis of rotation.

tau = r F sin(theta)

The radius of gyration K is the distance from the axis at which the whole mass of the body can be assumed to be concentrated so as to give the same moment of inertia: I = M K^2, hence K = sqrt(I / M). Its SI unit is the metre.

K = sqrt(I / M)