## Motion with Constant Acceleration

What sort of motion does this equation describe? x(t) = x_{0}+v

_{0}t + 1/2 a t

^{2}(1)

v(t) = v_{0}+a_{0}t (2)

a(t) = a_{0} (3)

This is motion with constant acceleration (a(t) = constant).

Equations (1) and (2) are important equations to remember, as problems with constant acceleration are very common in this course – for example, we will shortly see them when we start to talk about projectile motion in two dimension. There is a third important equation, which can be found thusly:

Rearranging (2) --> t=(v-v_{0})/a_{0} (4)

Substituting (4) into (1) yields:

- x=x

_{0}+ v

_{0}(v-v

_{0})/a

_{0}+ 1/2 a

_{0}(v-v

_{0})2/a

_{0}2

x=x_{0}+vv_{0}/a_{0} – v_{0}^{2}/a_{0} +1/2 v2/a_{0} –vv_{0}/a_{0} + v_{0}2/a_{0}

x=x_{0}+v^{2}/2a_{0} – v_{0}^{2}/2a_{0}

- v

^{2}-v

_{0}

^{2}=2a

_{0}(x-x

_{0}) (5)

Clearly, x=x_{0}+v_{average}t If we assume that vaverage = (v+v_{0})/2 (which is not generally true, but it is easy to show that it is true for constant acceleration), then we have:

- x-x

_{0}=v

_{average}(v-v

_{0})/a

_{0}

= (v+v_{0})(v-v_{0})/2a_{0} = (v^{2}-v_{0}^{2})/2a_{0}

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