Motion — Speed, Velocity and Acceleration

When we study motion, we measure many quantities such as how far an object goes, how fast it moves, and in which direction. These physical quantities are of two kinds, depending on whether or not direction matters. A scalar quantity has only magnitude (size) — a number with a unit — and no direction. A vector quantity has both magnitude and direction. Telling these apart is the first step in describing motion correctly.

Examples make the difference clear. Scalar quantities include distance, speed, mass, time, and temperature — for instance, saying "the distance is 5 km" or "the speed is 40 km/h" fully describes them without any direction. Vector quantities include displacement, velocity, force, and acceleration — for these we must state the direction too, for example "a displacement of 5 km toward the east" or "a velocity of 40 km/h northward". Direction is an essential part of a vector quantity.

Two important quantities in motion are distance and displacement, which are often confused. Distance is the total length of the path actually travelled by an object, regardless of direction; it is a scalar. Displacement is the shortest (straight-line) distance from the starting point to the final position, measured in a particular direction; it is a vector. If a person walks 3 km east and then 3 km back west to the starting point, the distance travelled is 6 km, but the displacement is zero, because they ended up where they started.

This distinction matters because distance and displacement can be very different for the same journey. Distance is always positive and adds up along the whole path, while displacement depends only on the start and end points and has a direction. For motion in a straight line without turning back, the distance and the magnitude of the displacement are equal. Understanding scalars and vectors, and the difference between distance and displacement, gives us the language we need to define speed, velocity, and acceleration in the topics that follow.

To describe how fast something moves, we use speed. Speed is the distance travelled by an object in unit time — in other words, how much distance it covers each second (or hour). Speed is calculated by the simple formula speed = distance ÷ time, written as v = d / t. The SI unit of speed is the metre per second (m/s), though we often use kilometres per hour (km/h) in daily life. Speed is a scalar quantity, since it tells us only how fast, not in which direction.

The motion of an object may be uniform or non-uniform. Uniform speed means the object covers equal distances in equal intervals of time — its speed stays the same throughout, as for a car on cruise control covering 20 m every second. Non-uniform speed means the object covers unequal distances in equal intervals of time — its speed keeps changing, as for a car in city traffic that speeds up and slows down. Most real journeys involve non-uniform speed.

For journeys where the speed changes, we use average speed, which is the total distance travelled divided by the total time taken: average speed = total distance ÷ total time. For example, if a car travels 100 km in 2 hours (even though its speed varied), its average speed is 100 ÷ 2 = 50 km/h. Average speed gives a single useful value to describe the overall journey, even when the actual speed was changing.

While speed tells us only how fast, velocity tells us how fast and in which direction an object moves. Velocity is the speed of an object in a particular direction — it is the rate of change of displacement. Because it includes direction, velocity is a vector quantity, while speed is a scalar. For example, "60 km/h" is a speed, but "60 km/h toward the north" is a velocity. Speed and velocity have the same units (m/s), but velocity carries the extra information of direction, which is important when studying motion that changes direction.

In real motion, the velocity of an object often changes — a car speeds up when the driver presses the accelerator, and slows down when the brakes are applied. The physical quantity that describes how quickly the velocity of an object changes is called acceleration. Acceleration is the rate of change of velocity — that is, the change in velocity in unit time. Because velocity is a vector, acceleration is also a vector quantity, having both magnitude and direction.

Acceleration is calculated by the formula acceleration = change in velocity ÷ time taken, written as a = (v − u) / t or a = Δv / t, where u is the initial velocity, v is the final velocity, and t is the time taken for the change. The SI unit of acceleration is the metre per second squared (m/s²). For example, if a car's velocity increases from 10 m/s to 30 m/s in 4 seconds, its acceleration is (30 − 10) ÷ 4 = 5 m/s².

Acceleration can be positive or negative. When the velocity increases (the object speeds up), the acceleration is positive. When the velocity decreases (the object slows down), the change in velocity is negative, so the acceleration is negative; this negative acceleration is also called retardation or deceleration. For example, a train slowing down as it enters a station has a negative acceleration (retardation).

An object has acceleration whenever its velocity changes, which can happen in three ways: when it speeds up, when it slows down, or even when it only changes direction at constant speed (since velocity includes direction). If the velocity does not change at all — the object is at rest or moving with constant velocity — the acceleration is zero. Understanding acceleration as the rate of change of velocity, and being able to calculate it, prepares us to interpret graphs of motion and to study Newton's laws, which connect force with acceleration.

The motion of an object can be shown clearly using graphs, which let us "see" how distance or speed changes with time. Two important graphs are the distance–time graph and the speed–time graph. Reading these graphs tells us about the object's speed, acceleration, and the distance it covers, often more easily than tables of numbers.

A distance–time graph has time on the horizontal axis and distance on the vertical axis. The key idea is that the slope (steepness) of the line represents the speed of the object. A straight, sloping line means the distance increases by equal amounts in equal times — that is, uniform speed (constant speed). A steeper line means a higher speed, while a gentler slope means a lower speed. A horizontal (flat) line means the distance is not changing, so the object is at rest. A curved line (getting steeper) shows that the speed is changing — non-uniform motion.

A speed–time graph has time on the horizontal axis and speed on the vertical axis, and it reveals the acceleration. A horizontal (flat) line means the speed is constant, so the acceleration is zero. A sloping straight line going up means the speed is increasing steadily, showing uniform (constant) acceleration; a line sloping down shows the object slowing (retardation). Importantly, the area under the speed–time graph gives the distance travelled by the object, which is a very useful result.

These two graphs are powerful tools. From a distance–time graph we read the speed (from the slope) and can tell whether the motion is uniform, non-uniform, or at rest. From a speed–time graph we read the acceleration (from the slope) and find the distance travelled (from the area under the line). Being able to draw and interpret graphs of motion is an important skill, allowing us to analyse and describe how objects move using simple lines and shapes.

The connection between force and motion was explained by Sir Isaac Newton in his three famous laws of motion. These laws describe how objects move when forces act on them and form the foundation of mechanics. Together they explain why objects stay still or keep moving, how force produces acceleration, and why forces always come in pairs.

Newton's First Law (the Law of Inertia) states that an object at rest stays at rest, and an object in motion keeps moving at a constant velocity (same speed and direction), unless an unbalanced (external) force acts on it. In other words, objects resist changes to their state of motion; this natural tendency to resist change is called inertia. A book lying on a table stays put until pushed; a ball rolling on a smooth floor would keep rolling forever if there were no friction. This is also why passengers lurch forward when a bus suddenly stops — their bodies tend to keep moving due to inertia.

Newton's Second Law tells us how much an object accelerates when a force acts on it. It states that the acceleration of an object is directly proportional to the applied force and in the direction of the force, and the relationship is given by the formula Force = mass × acceleration, or F = m a. From this, the SI unit of force, the newton (N), is defined as the force that gives a 1 kg mass an acceleration of 1 m/s². The second law shows that a larger force produces a larger acceleration, and that for the same force a heavier object accelerates less.

Newton's Third Law states that for every action, there is an equal and opposite reaction. This means that whenever one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first. These action and reaction forces act on different objects. Examples include a rocket that pushes gases downward and is pushed upward in return, a swimmer who pushes water backward and is pushed forward, and the recoil (kick) of a gun. Together, Newton's three laws — inertia, F = ma, and action–reaction — explain the motion of everything from a rolling ball to a launching rocket, completing our study of motion in this chapter.