Topic B: Force and Motion

Force and Motion is the core of mechanics: describing how things move and explaining why their motion changes. You will analyse motion using equations and graphs, add forces as vectors, apply Newton's three laws, use momentum to study collisions, treat projectile and circular motion, and connect everything through work, energy, power and gravitation.

Kinematics and motion graphs

Kinematics describes motion using displacement, velocity and acceleration. Velocity is the rate of change of displacement and acceleration is the rate of change of velocity. For uniform acceleration the equations of motion apply: v = u + a t, s = u t + one half a t squared, and v squared = u squared + 2 a s, where u is initial velocity, v is final velocity, a is acceleration, s is displacement and t is time. Motion graphs are powerful tools: on a displacement-time graph the gradient gives velocity, while on a velocity-time graph the gradient gives acceleration and the area under the line gives the displacement. Reading and sketching these graphs accurately is a frequent DSE skill.

Scalars, vectors and their addition

A scalar has magnitude only, such as distance, speed, mass, time and energy, while a vector has both magnitude and direction, such as displacement, velocity, acceleration, force and momentum. Vectors are added by drawing them head to tail or by using the parallelogram rule, and the single vector that replaces them is the resultant. A vector can also be resolved into two perpendicular components, typically horizontal and vertical, using F cos(angle) and F sin(angle). Resolving forces is essential for objects on slopes and for projectile motion, where the horizontal and vertical motions are treated independently. Always state both the size and the direction when giving a vector answer.

Newton's laws of motion and forces

Newton's first law states that an object stays at rest or moves at constant velocity unless acted on by a resultant force, a property called inertia. Newton's second law states that the resultant force equals mass times acceleration, F = m a, so a larger force gives a larger acceleration and a larger mass resists acceleration more. Newton's third law states that when body A exerts a force on body B, body B exerts an equal and opposite force on A; these action-reaction pairs act on different bodies. Common forces include weight (W = m g), the normal reaction, friction, tension and air resistance. Drawing a clear free-body diagram and finding the resultant force is the key to most mechanics problems.

Momentum and collisions

The momentum of an object is the product of its mass and velocity, p = m v, and it is a vector. Newton's second law can be written as force equals the rate of change of momentum, and the impulse F t equals the change in momentum. The principle of conservation of momentum states that in the absence of external forces the total momentum before a collision or explosion equals the total momentum after it. In an elastic collision kinetic energy is also conserved, whereas in an inelastic collision kinetic energy is lost (to heat, sound and deformation) although momentum is still conserved. These ideas explain recoil, car safety crumple zones, and rocket propulsion.

Projectile and uniform circular motion

A projectile moves under gravity alone once launched, and its horizontal and vertical motions are independent. Horizontally the velocity is constant because there is no horizontal force; vertically the object accelerates downward at g (about 10 m per s squared), so the path is a parabola. Uniform circular motion is motion in a circle at constant speed, but the velocity changes direction continuously, so there is a centripetal acceleration directed toward the centre. This requires a centripetal force F = m v squared over r, supplied by tension, gravity, friction or the normal force depending on the situation. There is no outward force; the apparent outward push is simply inertia carrying the object in a straight line.

Work, energy and power

Work is done when a force moves its point of application, and work done equals force times distance moved in the direction of the force, W = F s. Energy is the capacity to do work and is conserved: it changes form but is never created or destroyed. Kinetic energy is one half m v squared, and gravitational potential energy near the Earth is m g h. In the absence of friction the sum of kinetic and potential energy stays constant, which lets you solve many problems quickly, for example finding the speed of a falling object. Power is the rate of doing work or transferring energy, P = W over t, measured in watts. Real machines lose energy to friction, so efficiency, useful output over total input, is always less than one.

Gravitation

Newton's law of universal gravitation states that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres, F = G m1 m2 over r squared, where G is the universal gravitational constant. This single law explains the weight of objects on Earth, the orbits of the Moon and satellites, and the motion of planets. For a satellite in a circular orbit the gravitational force provides exactly the centripetal force needed, which fixes the relationship between orbital radius and period. A geostationary satellite orbits with a period of 24 hours above the equator so it stays above the same point on Earth.