Describing motion with calculus

Kinematics is the branch of mechanics that describes an object’s motion without considering the forces that cause it. Motion is usually modelled along a straight line, with positive and negative signs used to represent direction.

Displacement is a signed change in position, measured from a chosen origin and relative to a chosen positive direction. The sign matters. Negative displacement indicates motion or position in the direction defined as negative. The basic derivative relationship is

Velocity is the rate at which displacement changes with time. Since velocity is signed, its sign gives the direction of motion. Displacement increases when velocity is positive and decreases when velocity is negative.

Acceleration is the rate of change of velocity with respect to time. Therefore,

Velocity and acceleration don’t need to have the same sign. The object speeds up when their signs match and slows down when the signs are opposite.

These relationships connect three descriptions of the same motion:

• differentiate displacement to obtain velocity;
• differentiate velocity to obtain acceleration;
• integrate acceleration to obtain velocity;
• integrate velocity to obtain displacement.

An indefinite integral includes a constant. Use a known displacement or velocity at a stated time to find it. In practice, first write the general antiderivative, then substitute the given condition.

Gradients link the shapes of displacement-time, velocity-time and acceleration-time graphs. A stationary point on a displacement-time graph occurs when velocity is zero. On a velocity-time graph, a stationary point occurs when acceleration is zero.

Acceleration as a function of displacement

Velocity is sometimes given in terms of displacement rather than time. The chain rule then gives

This form is particularly useful when time doesn’t appear in the model. If velocity is written as a function of displacement, differentiate it with respect to displacement, then multiply by the velocity itself. Don’t use $dv/ds$ alone as the acceleration; its units aren’t those of acceleration.

Dot notation

A dot written above a variable is the conventional shorthand for differentiation with respect to time:

One dot represents one time derivative; two dots represent two time derivatives. Both mathematics and physics commonly use this notation.

Displacement from velocity

Integrating velocity over a time interval gives the change in displacement:

Geometrically, the integral represents the signed area between the velocity-time graph and the time axis. Areas above the axis make positive contributions, while areas below it make negative contributions. Parts corresponding to opposite directions may therefore cancel, which is correct when calculating displacement.

Speed and total distance

Speed is a scalar measure of motion equal to the magnitude of velocity. In symbols, speed is

It contains no direction.

Total distance travelled is the accumulated length of the path followed by an object, regardless of direction. It is calculated from

On a velocity-time graph, total distance equals the sum of the magnitudes of all areas between the graph and the time axis. A reliable method is to solve $v=0$ within the stated interval, split the integral at each relevant zero, then account for the sign on every subinterval. Alternatively, use technology to integrate the absolute-value function, provided it handles the turning points correctly.

Any time for which $v=0$ is an instant of rest. It marks a change of direction only when velocity changes sign. A zero that simply touches the time axis doesn’t necessarily show a reversal, so check the sign on either side.

In a complete kinematic problem, keep these distinctions clear:

• final displacement minus initial displacement gives net displacement;
• integrating velocity gives net displacement;
• integrating speed gives total distance;
• solving $v=0$ locates possible changes of direction;
• solving $a=0$ locates possible stationary points of the velocity function.

Mathematical conventions and wider connections

Kinematics creates a direct link with physics. Physics provides the interpretation of motion, while calculus provides the language of rates of change and accumulation. The same derivative and integral structures can describe quantities unrelated to physical motion, so kinematics belongs naturally within mathematics as well as mechanics.

Whether kinematics counts as central mathematics partly depends on educational and cultural choices about which applications receive emphasis. Mathematical knowledge has developed across many cultures. Curricula, universities and professional communities also shape which parts are classified as core mathematics. No single culture has sole ownership of the relationship between motion and mathematical change.

Convention matters too. Choosing a positive direction changes the signs of displacement, velocity and acceleration. Choosing $s$ instead of $x$ changes the notation, but neither choice alters the underlying motion. Dot notation works because mathematicians agree that a dot represents a time derivative. Similar conventions appear in other areas of knowledge: shared symbols allow communication, although agreement about a symbol isn’t evidence that the model itself accurately represents reality.