Topic 1: A. Space, time and motion

This theme builds the foundations of mechanics, describing how objects move and why their motion changes. You will quantify motion with vectors and graphs, then connect forces to acceleration, momentum and energy. The higher-level extensions reach into rotating rigid bodies and the strange behaviour of objects moving near the speed of light.

Kinematics and motion graphs

Kinematics describes motion without asking what causes it. Displacement is a vector measuring change in position, while distance is the scalar path length. The gradient of a displacement-time graph gives velocity, and the gradient of a velocity-time graph gives acceleration. The area under a velocity-time graph equals displacement. For uniform acceleration the SUVAT equations apply: v = u + a*t, s = u*t + 0.5*a*t^2, and v^2 = u^2 + 2*a*s. Worked example: a car starting from rest (u = 0) accelerates at 3 m/s^2 for 5 s. Final velocity v = 0 + 3*5 = 15 m/s. Distance s = 0*5 + 0.5*3*5^2 = 37.5 m. Always check that acceleration is constant before using SUVAT.

Projectile motion

A projectile moves under gravity alone, so its horizontal and vertical motions are independent. Horizontally the velocity is constant (no force), while vertically the object accelerates downward at g = 9.81 m/s^2. To analyse a launch, resolve the initial velocity into components: horizontal ux = u*cos(theta) and vertical uy = u*sin(theta). Worked example: a ball is kicked at 20 m/s at 30 degrees. uy = 20*sin(30) = 10 m/s; ux = 20*cos(30) = 17.3 m/s. Time to reach peak is when vertical velocity is zero: t = uy/g = 10/9.81 = 1.02 s. Total flight time (level ground) is twice this, 2.04 s, so horizontal range = ux*2.04 = 35.3 m. Air resistance is ignored unless stated.

Forces and free-body diagrams

A force is a push or pull, measured in newtons (N), and is a vector. A free-body diagram isolates one object and shows every force acting on it as an arrow from the object's centre. Common forces include weight (W = m*g, acting downward), normal reaction (perpendicular to a surface), tension, friction and drag. The net (resultant) force is the vector sum of all forces. If forces are perpendicular, combine them with Pythagoras. Worked example: a 4 kg box on a horizontal floor is pulled by a 12 N horizontal force against 5 N of friction. Net horizontal force = 12 - 5 = 7 N. Vertically, normal force balances weight, so N = m*g = 4*9.81 = 39.2 N. Always draw forces before writing equations.

Newton's three laws

Newton's first law states that an object continues at constant velocity (or stays at rest) unless acted on by a net external force; this defines inertia. The second law gives the quantitative link: net force equals mass times acceleration, F = m*a, with force and acceleration in the same direction. The third law states that for every action force there is an equal and opposite reaction force, acting on a different body. Worked example: applying the net force of 7 N from the previous box (mass 4 kg), acceleration a = F/m = 7/4 = 1.75 m/s^2. A common error is pairing the third-law forces on the same object; the reaction always acts on the other body in the interaction.

Momentum, impulse and conservation

Linear momentum is p = m*v, a vector with units kg*m/s. Newton's second law in its general form states that force equals the rate of change of momentum: F = (delta p)/(delta t). Impulse is the change in momentum, equal to force multiplied by time, J = F*t = delta p, with units N*s. In any collision or explosion with no external net force, total momentum is conserved. Worked example: a 2 kg trolley at 3 m/s collides and sticks to a stationary 1 kg trolley. Total momentum before = 2*3 + 1*0 = 6 kg*m/s. After, combined mass 3 kg moves at v: 3*v = 6, so v = 2 m/s. Kinetic energy is not conserved in this inelastic collision, but momentum always is.

Work, energy, power and efficiency

Work done equals force times displacement in the force's direction: W = F*s*cos(theta), measured in joules (J). The work-energy principle states that net work done equals change in kinetic energy, where Ek = 0.5*m*v^2. Gravitational potential energy near Earth is Ep = m*g*h. Energy is conserved overall, though some transfers to thermal energy through friction. Power is the rate of doing work, P = W/t, in watts (W), and also P = F*v. Efficiency is useful output energy divided by total input energy. Worked example: a motor delivers 600 J of useful output from 800 J of input, so efficiency = 600/800 = 0.75, or 75 percent.

Circular motion

An object moving in a circle at constant speed is still accelerating, because its velocity direction constantly changes. This centripetal acceleration points toward the centre and has magnitude a = v^2/r, where r is the radius. A net inward (centripetal) force is required: Fc = m*v^2/r. This force is provided by a real agent such as tension, friction or gravity, not a new force. Angular speed omega relates to linear speed by v = omega*r, and the period is T = 2*pi*r/v. Worked example: a 0.5 kg ball whirled on a 1.2 m string at 4 m/s needs Fc = 0.5*4^2/1.2 = 6.67 N of tension. There is no outward centrifugal force in this analysis.

Rigid body mechanics and torque (HL)

For higher level, rotation is analysed using torque (moment), the turning effect of a force: torque = F*r*sin(theta), in N*m, where r is the distance from the pivot. An object is in rotational equilibrium when the sum of clockwise torques equals the sum of anticlockwise torques. The rotational analogue of mass is the moment of inertia I, and the rotational form of Newton's second law is torque = I*alpha, where alpha is angular acceleration. Angular momentum L = I*omega is conserved when no external torque acts, explaining why a spinning skater speeds up when pulling their arms in. Worked example: a force of 20 N applied perpendicular at 0.3 m from a pivot gives torque = 20*0.3 = 6 N*m.

Special relativity basics (HL)

Special relativity rests on two postulates: the laws of physics are the same in all inertial frames, and the speed of light c is the same for all observers regardless of their motion. A consequence is that simultaneity, time and length are relative. The Lorentz factor gamma = 1/sqrt(1 - v^2/c^2) quantifies these effects. Moving clocks run slow (time dilation, t = gamma*t0) and moving objects contract along their motion (length contraction, L = L0/gamma). Worked example: a spacecraft moves at 0.6c, so gamma = 1/sqrt(1 - 0.36) = 1/sqrt(0.64) = 1.25. A 10 s interval measured on the ship is observed from Earth as 1.25*10 = 12.5 s. These effects are negligible at everyday speeds.