LC Applied Maths key terms

The change in position of a particle, measured as a vector from its starting point to its finishing point. It differs from distance travelled, which is a scalar and ignores direction.

The rate of change of displacement with respect to time; a vector quantity with magnitude (speed) and direction. Average velocity is total displacement divided by total time taken.

The rate of change of velocity with respect to time. Because velocity is a vector, a particle accelerates when its speed changes, its direction changes, or both change.

Motion in which the acceleration vector stays constant in both magnitude and direction. The five standard constant-acceleration formulae relate initial velocity, final velocity, acceleration, displacement and time.

A plot of velocity against time. Its gradient gives acceleration, and the area between the curve and the time axis gives displacement, with areas below the axis counted as negative.

The velocity of one body as observed from another moving body, found by vector subtraction of the observer's velocity from the body's velocity. Used for closest-approach and interception problems.

Splitting a vector into perpendicular components, usually horizontal and vertical, so that each direction can be treated independently. Essential for projectile and inclined-plane problems.

Motion of a particle projected into the air and subject only to gravity. Horizontal velocity stays constant while vertical velocity changes at the acceleration due to gravity.

The total time a projectile remains in the air between projection and return to a stated level. Found by solving the vertical displacement equation for time.

The horizontal distance a projectile travels before returning to its launch height. It is greatest for a launch angle of 45 degrees when air resistance is neglected.

The net force on a particle equals the rate of change of its momentum, which for constant mass reduces to force equals mass times acceleration. It links dynamics to kinematics.

A sketch showing a single body isolated with all external forces drawn as arrows acting at their points of application. Used to write correct equations of motion.

A contact force opposing relative motion or the tendency to move between surfaces. Limiting friction equals the coefficient of friction multiplied by the normal reaction.

The perpendicular contact force a surface exerts on a body resting on or pressing against it. It adjusts to the applied load and limits the available friction.

The turning effect of a force about a point, equal to force multiplied by the perpendicular distance from the point to the line of action. Measured in newton-metres.

For a body in equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments about that same point.

A pair of equal, opposite and parallel forces whose lines of action are separated, producing rotation without any net translational force. Its moment equals one force times the perpendicular separation.

The state in which the vector sum of all forces is zero and the sum of all moments about any point is zero, so the body has no translational or rotational acceleration.

The single point through which the total weight of a body may be considered to act, whatever its orientation. For a uniform body it coincides with the geometric centre.

Rotational failure of a body when the vertical line through its centre of gravity falls outside its base of support, so the weight creates an unbalanced moment about a base edge.

The state of a body on the point of moving, where friction has reached its maximum value and the applied forces are just balanced. Problems often ask for the critical angle or force.

The dimensionless ratio of limiting friction to the normal reaction between two surfaces. It depends on the materials in contact, not on the contact area.

The rate of change of angular displacement, usually measured in radians per second. It relates linear speed to radius by speed equals angular velocity times radius.

The acceleration of a body moving in a circle at constant speed, directed towards the centre. Its magnitude is the square of the speed divided by the radius.

The net inward force required to keep a body moving in a circular path, equal to mass times centripetal acceleration. It is provided by tension, gravity, friction or a normal reaction.

A mass on a string moving in a horizontal circle so the string sweeps a cone. The vertical component of tension balances weight while the horizontal component supplies centripetal force.

The time taken for one complete revolution, equal to the circumference divided by the speed, or two pi divided by the angular velocity. Its reciprocal is the frequency.

Oscillation in which acceleration is proportional to displacement from a fixed point and directed towards it. Described by sine or cosine functions with constant amplitude and period.

The maximum displacement of an oscillating particle from its equilibrium position. In simple harmonic motion it stays constant when damping is absent.

The progressive loss of energy from an oscillating system, reducing amplitude over time. Light damping allows oscillation; critical damping returns the system to equilibrium fastest without overshoot.

Every particle attracts every other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

The gravitational force per unit mass at a point, measured in newtons per kilogram. Near a planet's surface it is approximately constant; far away it falls with the inverse square of distance.

Motion of a body around a larger mass under gravity alone, where gravitational attraction provides the centripetal force. Circular orbits obey Kepler's third law relating period and radius.

Finding a derivative by forming the difference quotient of a small change in the function and taking its limit as the change tends to zero.

A rule for differentiating a composite function: differentiate the outer function, then multiply by the derivative of the inner function. Widely used with trigonometric and exponential functions.

A point on a curve where the gradient is zero. It may be a local maximum, a local minimum, or a point of inflection, classified using the second derivative or a sign test.

Problems where two quantities vary with time and are linked by an equation, so their rates are connected through the chain rule, such as the rate of change of volume and of radius.

The limit of a sum of areas under a curve between two limits, representing signed area. It is evaluated using an antiderivative and the fundamental theorem of calculus.

The region between a curve and an axis, computed as a definite integral. Regions below the axis contribute negative values, so sketches are needed for total area.

The solid formed by rotating a region about an axis, with volume found by integrating pi times the square of the radius over the limits.

The long-run average of a random variable, found by summing each outcome multiplied by its probability, or integrating value times density for continuous variables.

A measure of spread equal to the expected value of the squared deviation from the mean. Its square root is the standard deviation.

The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success. Defined by trial count and success probability.

A continuous symmetric bell-shaped distribution defined by its mean and variance. Many sample means approximate it, which underpins confidence intervals and hypothesis tests.

A procedure comparing sample data with a null hypothesis by calculating a test statistic and p-value. The null hypothesis is rejected when the p-value falls below the significance level.