Velocity

• 1. Introduction

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  Velocity is not the same thing as speed, and that distinction turns out to matter enormously. Physics draws a hard line between the two. Speed tells you how fast something is moving. Velocity tells you how fast and in which direction. A car cruising at a constant 20 kilometres per hour around a circular track is holding a steady speed the whole time. But because its direction keeps changing, it does not have a constant velocity. That means it is, technically, accelerating at every moment. That is the kind of puzzle velocity unlocks. It is a vector quantity, carrying both a magnitude and a direction inside a single measurement. And from that deceptively simple idea, a remarkable set of physical laws flows outward: the equations governing drag through a fluid, the minimum speed needed to break free of a planet, the way planets sweep across the sky in their orbits. The question the rest of this documentary takes up is what it really means to measure motion, and why that definition turns out to reach so far.

• 2. Speed Versus Direction

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  "5 metres per second east" is a velocity. "5 metres per second" is not. That one word, east, is the difference between a scalar and a vector. A scalar has magnitude alone. A vector carries direction too. In everyday speech, speed and velocity get used interchangeably, but in kinematics, the branch of classical mechanics devoted to describing the motion of physical objects, that casual swap creates real confusion. Velocity is measured in metres per second in the International System of Units, or SI. The same units apply to speed, because the scalar absolute value of a velocity vector is speed. But the vector form holds more information. To maintain a constant velocity, an object must hold both a constant speed and a constant direction simultaneously. Constant direction means straight-line motion. The moment a path curves, even if pace stays the same, velocity is changing, and change in velocity is exactly what acceleration means.

• 3. Average and Instantaneous

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  Position divided by time is not quite velocity. It is the change in position divided by the duration of a time interval, which gives average velocity over that stretch. That average is always less than or equal to average speed. The reason is that distance only ever increases, while displacement can shrink or reverse direction as an object doubles back on itself. On a displacement-versus-time graph, average velocity appears as the slope of a straight line connecting two points at the boundaries of the chosen time interval. Instantaneous velocity is something different and, at first, harder to accept. It is defined as the limit of average velocity as the time interval shrinks toward zero, which in calculus terms is the derivative of position with respect to time. One way to make that intuitive: instantaneous velocity is the speed and direction an object would keep travelling at forever, if it stopped accelerating at exactly that moment. On the same graph, instantaneous velocity is the slope of the tangent line touching the curve at a single point. The area under a velocity-versus-time graph, meanwhile, gives displacement; in calculus language, the integral of the velocity function is the displacement function.

• 4. Acceleration and the Suvat Equations

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  Acceleration is defined as the derivative of velocity with respect to time. Flip that around, and velocity is the area under an acceleration-versus-time graph, computed through integration. In the special case where acceleration is constant, a tidy family of equations called the suvat equations takes over. They link displacement, initial velocity, final velocity, acceleration, and time in ways that make many standard physics problems tractable. One result of working through these equations is the Torricelli equation, which relates velocity to displacement without needing time at all. The suvat equations are valid under both Newtonian mechanics and special relativity. Where those two frameworks part ways is in how observers at different speeds describe the same event. In Newtonian mechanics, all non-accelerating observers agree on the value of elapsed time and on measured acceleration. In special relativity, neither of those agreements holds. Only relative velocity can be calculated, and its value depends on the reference frame chosen.

• 5. Quantities That Depend on Velocity

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  Momentum in classical mechanics is defined by Newton's second law as the product of an object's mass and its velocity vector. Because velocity is a vector, momentum is a vector too. Kinetic energy is built differently. It depends on the square of the velocity, which strips the directional information away and makes it a scalar quantity. Drag, the force that a fluid exerts against a moving object, is also tied to velocity squared. Its formula involves the fluid's density, the object's cross-sectional area, and a dimensionless drag coefficient. Doubling an object's speed quadruples the drag force working against it. Escape velocity is another quantity rooted in velocity, though the name is a slight misnomer. The escape velocity from Earth's surface is about 11,200 metres per second. It is independent of direction: any object reaching that speed, regardless of which way it is pointed, will leave Earth's gravitational grip as long as nothing else blocks its path. That independence from direction is precisely why "escape speed" would be the more accurate label. In special relativity, the Lorentz factor, denoted by the Greek letter gamma, appears repeatedly in equations that describe how measurements change at speeds approaching the speed of light, designated c.

• 6. Relative Velocity and Reference Frames

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  Relative velocity measures the motion of one object as seen from the frame of another. If object A moves with velocity vector v and object B moves with velocity vector w, the velocity of A relative to B is simply v minus w. In Newtonian mechanics, that result is the same no matter which inertial reference frame an observer chooses. Special relativity breaks that rule. Velocities become frame-dependent, and the simple subtraction no longer applies at speeds close to the speed of light. In one dimension, the arithmetic simplifies further. Two objects moving in the same direction produce a relative speed equal to the difference of their individual speeds. Two objects moving in opposite directions produce a relative speed equal to their sum.

• 7. Coordinate Systems

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  In a two-dimensional Cartesian system, velocity splits into components along the x-axis and the y-axis. The magnitude of that two-dimensional velocity vector, which equals speed, is recovered using the distance formula: the square root of the sum of the squared components. A third axis adds a z-component, and the formula extends in the same pattern. Polar coordinates offer a different decomposition. There, a two-dimensional velocity breaks into a radial component, pointing toward or away from the origin, and a transverse component, perpendicular to that radial direction. Both of these arise from angular velocity, the rate of rotation about the origin. In a right-handed coordinate system, counter-clockwise rotation carries a positive sign and clockwise a negative one. Transverse speed equals the product of angular speed and the radius from the origin. Angular momentum in scalar form is the mass times the distance to the origin times that transverse speed. When the only forces acting on an object point radially inward and fall off with the square of distance, as with gravity in an orbit, angular momentum stays constant throughout the motion. The rate at which the orbit sweeps out area also stays constant. Those relationships are called Kepler's laws of planetary motion.