Acceleration Fundamentals

Acceleration is the rate of change of velocity. Velocity has magnitude and direction, so turning at constant speed still involves acceleration. In a chosen Cartesian direction, instantaneous velocity is v = dx/dt and acceleration is a = dv/dt. The SI unit of acceleration is m/s².

Over a finite interval, average acceleration is (v₂ − v₁)/(t₂ − t₁), with t₂ greater than t₁. It does not determine the acceleration at either endpoint or the peak between them. These distinctions follow NASA's displacement, velocity and acceleration definitions.

Choose the positive direction before assigning signs. Negative acceleration means acceleration in the negative direction. It slows an object moving positively but speeds up an object already moving negatively.

Constructed comparison: the same velocity change of +20 m/s is achieved over four different elapsed times. Each row describes a separate interval, not successive instants in one journey.

Doubling the duration halves the average for the same velocity change. This inverse relationship is not exponential decay. Nor does it establish an initial acceleration: many different acceleration histories have the same velocity change and elapsed time.

For constant acceleration along a fixed axis, measured from time zero: v = v₀ + at and x − x₀ = v₀t + ½at². Integrating the velocity gives displacement, which can differ from total distance travelled if motion reverses. NASA's ballistic-motion derivation applies these relationships to a constant gravitational acceleration with drag omitted.

Constructed braking example: v₀ = +20 m/s and a = −4 m/s² until rest. The stopping time is 5 s and displacement is 20 × 5 − ½ × 4 × 5² = 50 m. These are assumed inputs, not a recommended braking capability or road stopping-distance standard.

Do not extend that braking model past 5 s without changing the assumptions: constant negative acceleration would then predict motion in reverse. Driver reaction time, changing traction and actuator response are absent from this example.

When acceleration varies, use Δv = ∫a(t)dt and displacement = ∫v(t)dt over the stated interval. A single average acceleration generally cannot replace the time history in the displacement equation.

A fixed probe measures changes at one location. A moving fluid particle also encounters spatial changes in velocity. In Cartesian coordinates its acceleration is Dv/Dt = ∂v/∂t + (v · ∇)v, the local term plus the convective term. MIT's hydrodynamics notes, page 8 derive this material derivative.

For a one-dimensional velocity field u(x,t), a = ∂u/∂t + u ∂u/∂x. As a constructed kinematic example, let steady u(x) = 2 m/s + (0.5 s⁻¹)x. At x = 4 m, u = 4 m/s and particle acceleration is 4 × 0.5 = 2 m/s² although the local time derivative is zero.

This prescribed field illustrates the derivative; it is not a solved pipe network. A physical flow must also satisfy mass conservation and momentum balance. For example, a steady incompressible flow with increasing mean speed needs decreasing cross-sectional area at fixed volumetric flow.

For a constant-mass particle in an inertial frame, net force is ma. Include all relevant forces, rather than treating acceleration as the contribution of one chosen force. See the gravity and Newton's-law worked examples.

Comfort, machinery startup and impact assessment need a defined time history and application-specific criteria. This article supplies no universal elevator-comfort threshold, emergency-braking limit or safety factor. Angular acceleration describes changing angular velocity and uses rad/s²; it is not interchangeable with linear acceleration.

The pump-control guide considers operating points. The pump-duty calculator evaluates steady conditions and does not predict startup pressure surges or water hammer.