Chapter 7: Mechanics, Probability, and Statistics

Kinematics

For motion in a straight line with constant acceleration:

For variable acceleration, use calculus: , .

Projectiles (launch speed at angle , no air resistance, acceleration downward): horizontal motion is uniform and vertical motion is uniformly accelerated. Then

Example: , , : ; ; .

Dynamics

Newton’s laws: (1) a body remains at rest or in uniform motion unless acted on by a net force; (2) ; (3) forces between two bodies are equal and opposite. For connected bodies, apply to each body and solve the equations together. Friction: where is the normal reaction; limiting friction is . On a smooth incline of angle , (so at with ).

Work, energy, and power: ; kinetic energy ; gravitational potential energy ; power . With no non-conservative forces, mechanical energy is conserved; with friction, the work–energy theorem gives .

Momentum and impulse: ; impulse . In a collision, momentum is conserved; Newton’s experimental law states that (speed of separation) (speed of approach), where is the coefficient of restitution. Example: a 2 kg body moving at hits a 1 kg body at rest and they stick: ; kinetic energy falls from 9 J to 6 J, so 3 J is lost.

Circular motion and SHM. For uniform circular motion, the acceleration is towards the centre. In simple harmonic motion, , with solution , period and speed (Halliday et al., 2018).

Statics

A body is in equilibrium when the resultant force is zero and the resultant moment about any point is zero. The moment of a force about a point is force × perpendicular distance. Typical problems: ladders against walls, beams on supports, bodies on rough inclines, and the position of a centre of mass. Use a clear diagram, resolve forces horizontally and vertically, and take moments about a convenient point.

Probability

For events and : ; conditional probability ; events are independent if . Bayes’ theorem:

Counting: permutations ; combinations .

Worked example (Bayes). A disease affects 1% of a population. A test detects it in 95% of those who have it and gives a false positive in 10% of those who do not. If a person tests positive, , about 8.8%. Most positives are false, because the disease is rare. Understanding this is vital in medicine and engineering reliability.

Random variables and distributions

For a discrete random variable : and . The binomial distribution counts successes in independent trials: with , mean , variance . Example: , : . The normal distribution is a continuous bell-shaped model; a value is standardized by , the same quantity as the Z-score in Chapter 2 (Walpole et al., 2012).

Descriptive statistics

Measures of centre: mean , median, mode. Measures of spread: range, interquartile range, variance , and standard deviation . Example: data 2, 4, 4, 4, 5, 5, 7, 9 have , , , .

Common mistakes

Using the constant-acceleration formulae when acceleration varies.

Omitting friction’s direction (it opposes motion or tendency to move).

Treating and as the same.

Forgetting that energy is a scalar and momentum is a vector.

Practice questions

A car accelerates from rest at for 10 s. Find the distance travelled. [100 m]

A 5 kg block is pulled across rough ground by a 30 N horizontal force with and . Find its acceleration. [2 m s⁻²]

Find when . [15/16]

Show that .

Part Three: Physics

Physics explains how the natural world works, using a small number of laws expressed in mathematics. The A/L syllabus covers measurement, mechanics, oscillations and waves, thermal physics, properties of matter, electricity and magnetism, electronics, and modern physics. The chapters below follow that sequence. Fuller treatments are in standard texts such as Halliday et al. (2018), Young and Freedman (2020), Serway and Jewett (2019), and Giancoli (2014). Numerical values of constants are in Appendix B and follow the international recommended values (Tiesinga et al., 2021).