CIE International A-Level Maths: Mechanics

Differentiate vectors from scalars in kinematics by using displacement and distance, and velocity and speed, to show direction and negative values with standard units.

Learn how displacement-time graphs encode velocity through gradient changes. Identify forward and backward motion, stationary periods, instantaneous rest, negative displacement, and curved graphs.

Explore a curved displacement-time graph for a point on machinery moving vertically, calculating the greatest displacement, greatest distance from start, the interval of downward movement, and instants of rest.

Analyze velocity-time graphs by distinguishing velocity from speed, interpreting acceleration as the gradient, and using the area under the curve to compute displacement, with sign indicating forward or backward motion.

Analyze velocity-time graphs to compute distance and displacement from areas under the curve, using triangles and trapeziums, and distinguish distance traveled from displacement.

Sketch a velocity-time graph for a runner starting from rest, accelerating to 8 m/s in five seconds, then maintaining speed for t seconds and decelerating to stop.

Explore acceleration-time graphs: area under the curve gives velocity, and the gradient shows the rate of change of acceleration; learn to read values and distinguish speeding up from slowing down.

Explore the Suvat equations of constant acceleration, define s, u, v, a, t, and learn the five formulas with a practical example showing how to solve for missing values.

Explore the five Suvat equations with a velocity-time graph and area reasoning to derive v = u + a t and s = u t + 1/2 a t^2.

Explore SUVAT equations in a constant-acceleration motion on a straight runway, computing acceleration, time, and midpoint speed from a worked aircraft example.

apply SUVAT equations with constant acceleration to find the time for 100 m from 5 m/s with 0.5 m/s²; a Casio ClassWiz calculator shows two roots; select 12.36 s.

Learn how to solve two-part journey problems by setting up two suvat systems for each leg, using a common velocity and constant acceleration to form and solve simultaneous equations.

Use suvat equations to compare moving objects, form a simultaneous equation for Car A and Car B on a straight road, and find the overtaking time t = 5 s.

Explore the gravitational constant G and its role as Earth's acceleration. Learn why we use values like 10, 9.8, or 9.81 in Cambridge international a-level maths exams, noting altitude variations.

Explore projectiles under gravity while ignoring air resistance. Use s = ut + 1/2 at^2 with g ≈ 9.8 m/s^2, define a positive direction, and compute time and final velocity.

Explore a projectile example using the particle model under gravity with no air resistance; find max height 60 m, impact speed 34.3 m/s, and total time 5 s.

Solve a projectile problem by treating down as positive, using s = ut + 1/2 a t^2 to find h and v = u + at for impact speed.

Master force diagrams in dynamics by representing all acting forces as vectors, including weight and the chair's reaction, and applying Newton's laws to analyze interactions.

Create force diagrams for book-on-rough-table and lift scenarios from two perspectives, highlighting weight, reaction forces, friction, tension, and Newton's third law.

Identify the resultant forces as the net effect of several forces on an object by summing horizontal and vertical components, then use the zero resultant criterion to check equilibrium.

Model forces as vectors and apply nose-to-tail addition to find the resultant; magnitude is ten newtons and direction is given by the angle to horizontal or bearing.

Represent resultant forces as vectors with i and j components, compute magnitude by Pythagoras, and determine bearings from north, illustrated by the -6i + aj, magnitude 8 example.

Explore how forces in column vector form simplify finding the resultant, its magnitude via Pythagoras, and the angle with the horizontal using tan theta, with example calculations in newtons.

Determine parallel forces by expressing q as k(i+j) and the resultant as p(i+2j). Equate components to solve for k and find |q| = 10√2 newtons.

Demonstrates Newton's second law, F = m a, linking resultant force to mass and acceleration, and uses simple horizontal-force examples to show calculating acceleration from net force.

Apply Newton's second law, f = ma, to a 600 kg car under braking to determine the braking force B from the given deceleration.

Explore the difference between mass and weight, derive weight as m g via Newton's second law, and solve a lift problem with 3000 N tension and 0.5 m/s² downward acceleration.

Link forces with kinematics by deriving acceleration from f=ma and applying suvat equations to solve motion problems, including a box and a braking car.

Analyze connected particles in a car‑and‑trailer system using forces, f=ma, and simultaneous equations; derive acceleration and tow‑rope tension, then apply suvat to a rope‑break scenario.

Analyze pulley problems by drawing force diagrams for each mass, apply f=ma with consistent directions, solve simultaneous equations for acceleration and tension, and note the role of g.

Explore connected particles over a smooth fixed pulley, applying force diagrams and Newton's second law to form and solve simultaneous equations for tension and acceleration at one quarter g.

Tackle harder connected particles problems involving changing forces where the motion changes mid-question, compute acceleration and tension, then determine the resistive force and subsequent distance after the string goes slack.

Analyze a two-bucket pulley problem with 4 kg and 6 kg. Use acceleration 1.96 m/s² and v^2 = u^2 + 2as to find maximum height of p at 4.4 m.

Explore modeling assumptions in dynamics, including objects as particles, light inextensible strings, and smooth pulleys, to justify equal tensions and accelerations in mechanics problems.

Learn how acceleration varies with time and how displacement and velocity become functions of time, using velocity-time graphs, area under curves, and solving s(t) and v(t) equations.

Explore how differentiation and integration link displacement, velocity, and acceleration through velocity-time graphs; differentiate to obtain acceleration and integrate velocity to get displacement from the area under the curve.

Differentiate the displacement x = p t^2 + q t + 5 to obtain velocity and acceleration, determine p and q from t=1, and find x(2)=11 m (position, not distance).

Differentiate displacement to find turning points, yielding max displacement of five meters in the first two seconds and maximum velocity of 16.25, found by setting acceleration to zero.

Integrate velocity to obtain displacement and thus position, undo differentiation in the process, apply x(0)=10, and evaluate at t=3 to get x=46 m; note this is position, not distance traveled.

Apply integration to mechanics questions: derive velocity from acceleration, locate the next rest time, sketch the velocity-time graph, and compute distance traveled in five seconds as 13 m.

Explore how the Suvat equations arise from differentiating s = u t + 1/2 a t^2 under constant acceleration, yielding v = u + a t and a = dv/dt.

Learn to use a calculator to speed up mechanics in maths A-level exams, solving a polynomial and evaluating definite integrals quickly with a Casio Classwiz, saving time for more marks.

Resolve forces into horizontal and vertical components, and into parallel and perpendicular components on a slope to find the resultant force, illustrated with tension, weight, and the normal force.

Apply forces, resultant forces, and components with Newton's second law to predict motion and calculate acceleration for objects like a sledge or boat.

Explore how to resolve forces on an inclined plane using parallel and perpendicular components to determine acceleration and motion, with suvat-based examples.

Apply weight components and friction on an inclined plane to determine the resultant force. Use Newton's laws and suvat relations to find friction and tension in rough-plane problems.

Explore how friction is calculated from the coefficient of friction and the reaction (normal) force, then determine motion or acceleration for forces on horizontal and inclined surfaces.

Analyze friction on a 30 degrees incline by resolving weight into components, calculating the normal force and friction, then applying Newton's second law to obtain 3.2 m/s^2 down the slope.

Compute the frictional force for a four kilogram particle on a rough incline with a 45 N rope, mu 0.3, and 30 degrees by resolving forces and checking up-slope movement.

Examine dynamics on a rough 20-degree incline with friction mu=0.3, deriving the up-slope distance and ascent and return times using weight, friction, and components.

Resolve forces on a 6 kg particle on a 25° plane with a 12 N force parallel to the slope and 0.7 m/s² acceleration; compute friction and reaction, then mu.

Resolve the weight and a horizontal 12 N force into components on a rough 25° incline, compute the normal reaction, then apply f = ma to find mu ≈ 0.654.

Calculate the horizontal force to push a 4 kg particle up a rough 15-degree incline with mu 0.45 and acceleration 0.25 m/s^2, using parallel and perpendicular forces and friction.

Analyze two masses linked by a string over a pulley on a rough 30-degree incline, derive the tension by applying Newton's laws and solving simultaneous equations.

Analyze a system of connected particles on a rough and smooth 30-degree inclined plane with a pulley, deriving the string tension, friction coefficient mu, and the pulley force.

Explore statics and equilibrium by splitting forces into horizontal and vertical components, using up equals down and left equals right, to solve three-force and string-tension problems with practical friction context.

Apply the triangle method to solve three-forces equilibrium problems by nose-to-tail vector addition, turning force systems into a simple triangle and trig calculations (sine rule, cosine rule).

Resolve forces on a six-kilogram particle on a 30-degree incline into parallel and perpendicular components, then find F and R, with F ≈ 33.9 N and R ≈ 67.9 N.

Analyze connected particles in equilibrium on an inclined plane by resolving forces into parallel and perpendicular components, determining the tension, the required horizontal force, and the normal reaction.

Examine statics of hanging objects in equilibrium by resolving tensions and weights into horizontal and vertical components. A smooth bead keeps tension equal on both sides.

Explore limiting equilibrium and friction, showing how horizontal and angled pushes determine if an 8 kg object stays at rest or starts moving under mu r.

Evaluate limiting equilibrium on a rough 30-degree slope to find the rope tension for up and down sliding. Resolve forces into parallel and perpendicular components and apply friction mu r.

Explore a limiting-equilibrium problem on a rough plane at a 30-degree incline. Determine p and mu using weight 10 N and normal reaction 18 N.

Define work done as energy transferred when a force acts across a distance, measured in joules. Use the formula work equals force times distance, with gravity and friction examples.

Resolve a force into its components and take the component acting in the direction of motion. Multiply that component by distance to find work, including friction and gravity on slopes.

Derive the kinetic energy formula KE = 1/2 m v^2 from motion equations, and illustrate how speed determines energy with examples of a ball and a slope.

Gravitational potential energy is the energy stored by height, defined as m g h. For a 1600 kg car rising 72 m in 30 s, the change is 1,152 kJ.

Apply force analysis and energy methods to gravitational potential energy problems on inclined planes with friction. Derive acceleration and potential energy loss, and solve a hill-car work-energy problem.

Apply the work energy principle to relate changes in mechanical energy, kinetic plus gravitational potential, to the work done by forces including friction and air resistance, with practical problem examples.

Apply the work-energy principle to incline and rough-plane problems, calculating gravitational potential energy changes, friction work, and final speeds using energy methods.

define power as the rate of energy transfer and use the force–velocity formula to solve problems, including engine output, kinetic energy, and maximum speed under resistance.

Explore power with practical problems linking acceleration, speed, and energy. Compute changes in kinetic and gravitational potential energy, work against resistance, and engine power in hill and level scenarios.

Momentum equals mass times velocity, describing mass in motion and how heavier or faster objects resist change; the lecture uses simple problems and unit analysis to illustrate momentum and vectors.

Explore the conservation of linear momentum on a frictionless surface, using before and after state diagrams to solve collision problems, including rebounds, coalescence, and mass ratios.

Explore conservation of linear momentum in multi-collision problems with moving spheres, solving sequential collisions where A and B coalesce, then collide with C, to find final speeds.