Acceleration - Definition and Equations

The acceleration of a mass point is defined on Page 1. Two sets of basic equations for the products of acceleration and time elapsed and acceleration and displacement are stated below. The numbering of equations continues from Page 1.

Acceleration Multiplied By Time Elapsed

Treating the symbol dvector v/dt in Eq.(1) for the acceleration vector a as a ratio of two infinitesimal quantities and solving for dvector v gives

vector adt = dvector v. space(3)

Eq.(3) applies to an infinitesimal time interval of duration dt. It equates the product of the particle's acceleration and the infinitesimal time elapsed dt to the change dvector v in the particle's velocity during the elapsed time interval.

To find the change Δvector v in the particle's velocity during a finite time interval of duration Δt, one has to sum the terms vector adt for all infinitely many infinitesimal time intervals that make up Δt. A sum over infinitesimals is called an integral. Calculus provides rules for working out integrals. One needs to know how the acceleration depends on time in order to work out the integral for a specific case.

For the special case in which the acceleration is constant during a time interval of finite duration Δt, the integral (sum) of the vector adt terms can be done without knowing the rules of calculus. The result is

vector aΔt = Δvector v, spaceif vector a = const.space(4)

Acceleration Multiplied by Displacement

The infinitesimal displacement dvector r of a particle during an infinitesimal time interval is a vector quantity. Multiplying the acceleration of a particle by the particle's infinitesimal displacement means using the scalar product. The result is:

vector aScalar Productdvector r = (1/2)dv2 spaceor space2vector aScalar Productdvector r = dv2 space (5)

where v is the particle's speed.

For the special case in which the acceleration is constant during a finite time interval while the particle undergoes the finite displacement Δvector r, the terms vector aScalar Productdvector r can be summed (integrated) without knowing any formal rules of calculus. The result is:

vector aScalar ProductΔvector r = (1/2)Δv2 spaceorspace2vector aScalar ProductΔvector r = Δv2,space if vector a = const.space(6)

For one-dimensional motion along an x-axis, Eqs.(5) and (6) reduce to the following Eqs.(7) and (8), respectively.

2axdx = dv2space(7)

2axΔx = Δv2,spaceif vector a = const.space(8)

For a derivation of Eqs.(5) and (7), see Acceleration/Instantaneous/Explain It.

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