Acceleration

• 1. Introduction

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  Acceleration shapes nearly every physical experience a person has, yet most people never think of it by name. An elevator pressing you into the floor. A spacecraft firing its retrorockets to slow down. A satellite silently curving around the Earth without changing speed at all. Each of these is acceleration at work. In physics, acceleration is defined as the rate of change of velocity. That single definition hides a great deal. Velocity is not just speed; it is speed in a particular direction. So acceleration covers any change in how fast something moves, yes, but also any change in which way it is moving. A car rounding a corner at constant speed is accelerating. A falling object gaining speed straight downward is accelerating. Both fit the same definition, and yet they feel entirely different to anyone inside them. Acceleration is also a vector quantity, meaning it carries both a magnitude and a direction. The standard unit used to measure it is the metre per second squared. What remains to be understood is how acceleration divides into distinct types, how Newton's laws tie it to force, and what happens when speeds approach that of light.

• 2. Two Kinds of Measurement

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  Average acceleration is the most straightforward form of measurement. It requires nothing more than knowledge of the change in velocity and the change in time over a given period. In a strict sense, it is the only true acceleration one can measure directly without appealing to an empirical law, making it the most fundamental form of acceleration measurement. For short intervals, physicists often assume that acceleration is uniform, meaning the object changes velocity by an equal amount in every equal stretch of time. Under that assumption, average acceleration and the exact moment-to-moment value coincide. Instantaneous acceleration, by contrast, is the limit of the average acceleration over an infinitesimally small interval of time. In the language of calculus, it is the derivative of the velocity vector with respect to time. Because velocity is itself the derivative of position with respect to time, instantaneous acceleration can also be read as the second derivative of position. By the fundamental theorem of calculus, the area under a curve on an acceleration-versus-time graph corresponds exactly to the change in velocity over that interval. One step further, the derivative of the acceleration function is called the jerk function, and its integral reveals how acceleration itself changes at a given moment.

• 3. Force, Mass, and Newton

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  Newton's second law of motion sits at the heart of why acceleration happens. For a body with constant mass, the vector acceleration of the body's centre of mass is proportional to the net force acting on it. The direction of the net acceleration matches the direction of the net force, and the magnitude of that acceleration grows with the force but shrinks as mass increases. A measurement of average acceleration, then, is also indirectly a measurement of the average force on the object, a quantity also known as impulse. In an inertial reference frame, this relationship is clean and direct. In a reference frame that is itself accelerating, Newton's laws can still be applied, but only by introducing what is called an inertial force, or fictitious force, on the mass, pointing opposite to the acceleration of the frame. This fictitious force accounts for the tendency of any mass to maintain its existing state of motion. A person riding in an elevator feels heavier as the elevator accelerates upward and lighter as it decelerates. That sensation is the fictitious force made tangible. In fact, a mechanical accelerometer works on exactly this principle: if the acceleration of the frame is known, measuring the supporting force on a mass inside it can be used to infer the acceleration.

• 4. Tangential and Radial Components

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  When an object travels along a curved path, its acceleration splits into two perpendicular pieces. The tangential acceleration is the component aligned with the direction of motion. It governs changes in speed. The radial acceleration, also called normal or centripetal acceleration, is the component pointing inward toward the centre of curvature. It governs changes in direction. Passengers in a vehicle feel these components as distinct sensations. The tangential component, when the vehicle is speeding up or slowing down, pushes them back into their seats or forward against their restraints. The radial component, when the vehicle turns, pushes them toward the outside of the curve. That outward sensation is the reaction to centripetal acceleration; in the body's reference frame it appears as a centrifugal force, another fictitious force arising from the body's linear momentum as it tends to continue in a straight line. In uniform circular motion, where speed is constant, all acceleration is radial. The magnitude of centripetal acceleration for a given speed is inversely proportional to the radius of the circle, and grows as the square of the speed. For a given angular velocity, the centripetal acceleration is directly proportional to the radius, because velocity itself depends on radius. The geometric analysis of three-dimensional curved paths, including the tangent, principal normal, and binormal directions, is described by the Frenet-Serret formulas.

• 5. Free Fall and Parabolic Paths

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  Free fall is one of the most cited examples of uniform acceleration. When a body falls in a uniform gravitational field with no resistances to motion, the only factor governing its acceleration is the gravitational field strength, also called acceleration due to gravity. Because that value is constant, the simple formulas relating displacement, initial velocity, time-dependent velocity, and elapsed time all apply. Galileo demonstrated that motion under constant acceleration can be resolved into two orthogonal parts: one at constant velocity and one governed by uniform acceleration. The net result is parabolic motion. This is the trajectory a projectile follows in vacuum near the surface of Earth. Deceleration, which is sometimes called retardation, is not treated as a separate category in Newtonian mechanics. It is simply acceleration directed opposite to the velocity vector. In spacecraft, such deceleration is often achieved by retrorocket burning, and passengers or payloads experience the inertial force as a push in the direction of original travel. Both acceleration and deceleration continue to be felt until the relative velocity between the passenger and the vehicle reaches zero.

• 6. Relativity and the Speed of Light

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  Albert Einstein identified a profound consequence of gravity and inertial acceleration. Unless the state of motion of an object is known, it is impossible to distinguish whether an observed force comes from gravity or from acceleration. Einstein called this the equivalence principle. He concluded that only observers who feel no force whatsoever, including no gravitational force, are justified in concluding they are not accelerating. This insight underpins general relativity, where gravity and inertial acceleration may be locally indistinguishable. Special relativity, meanwhile, describes how the classical equations break down at speeds approaching that of light in vacuum. Newtonian mechanics remains a valid approximation at lower speeds, but as speeds climb toward the speed of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached. An object with mass can approach the speed of light asymptotically but can never reach it. The accelerometer, which measures proper acceleration relative to free fall, remains the instrument of choice for measuring acceleration without reference to any external frame, and its readings in a strong gravitational field connect directly to the equivalence principle Einstein described.