Derivations

Kinematic equations

Derive the three specified equations of motion for constant acceleration in a straight line.

• Level: Higher Level derivation.
• The resulting equations may be used only when acceleration is constant and motion is in a straight line.
• u = initial velocity, v = final velocity, a = acceleration, s = displacement and t = time.

Related equations: s = ½(u + v)t follows from displacement = average velocity × time. Substituting u = v − at gives s = vt − ½at².

Resistors in series and parallel

Derive the total-resistance formulae from conservation of energy, conservation of charge and Ohm’s law.

• Level: deriving the formulae is Higher Level.
• Using the series and parallel resistance formulae is required at both Ordinary Level and Higher Level.
• Rₜ denotes the total or equivalent resistance.

Reasonableness check: total resistance increases when resistors are added in series. In parallel, the total resistance is less than the smallest branch resistance.

Newton’s second law: F = ma

Derive F = ma for constant mass from force as the rate of change of momentum.

• Supporting derivation from the previous course notes; useful for understanding, but not listed with the two explicitly mandatory derivation groups in the 2025 course specification.
• Assume the mass is constant.

Linear and angular speed: v = rω

Use the definition of the radian to connect linear speed and angular speed.

• Supporting derivation; θ must be measured in radians.

Orbital period and radius

Derive T² = 4π²r³/(GM) for a circular orbit.

• Supporting derivation for a circular orbit about a central mass M. The orbital radius is measured from the centre of M.

Meaning: for bodies orbiting the same central mass, 4π²/(GM) is constant, so T² ∝ r³.

Diffraction grating condition

Use path difference to obtain nλ = d sin θ for principal maxima.

• Supporting geometrical derivation. d is the grating spacing; θ is measured from the normal.

Common error: the angle is measured from the grating normal, not from the plane of the grating.

Force on a moving charge: F = Bqv

Derive the magnetic force on one charge from the motor-force equation.

• Supporting derivation for motion perpendicular to a uniform magnetic field. More generally, F = Bqv (velocity perpendicular to the field).

For a negative charge, the force direction is opposite to the direction predicted for conventional positive charge.