The equations of kinematics for constant acceleration

FIGURE 2.9

When the acceleration is constant, its average and instantaneous values are identical, so we may write a = (υ _f − υ _i)/(t _f − t _i). When we set t _i = 0 and t _f = t, and use the new notation, we obtain a =(υ − υ ₀)/ t, or

υ = υ ₀ + at (2.9)

FIGURE 2.10 The x versus t graph for constant (positive) acceleration is a parabola. The slope of the tangent at t = 0 is the initial velocity υ ₀.

On a υ versus t graph, this is the equation of a straight line whose slope is a (see Fig. 2.9). With t as the elapsed time, the displacement ∆ x = x − x ₀ is given by Eq.2.8:

(2.10)

If Eq.2.9 for υ is used in Eq.2.10, we find

(2.11)

This is the equation of a parabola, as shown in Fig. 2.10. We can eliminate time from Eq.2.11 by using t = (υ − υ ₀)/ a from Eq.2.9. After some rearrangement we obtain

(2.12)

Equations 2.9 – 2.12 are the equations of kinematics for acceleration that is con­stant both in magnitude and in direction along the x axis. Of the five quantities x, υ ₀, υ, a, and t, we need to be given at least three to solve any problem. The value of x ₀ depends on where the origin is placed. We usually try to have x ₀ = 0 to simplify the problem.

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