8Bantidiff


 * Kinematics using Antidifferentiation**

math \\ \text{Since } . \qquad a=\dot{v}=\ddot{x} \\. \\ \text{It follows that } math math \\ . \qquad v=\displaystyle{\int a \;dt} \\. \\ \text{and} \\ . \qquad x= \displaystyle{\int v \;dt} math

Example

A particle, initially at rest at x = 2 has an acceleration given by: math . \qquad a=12 \,\cos(2t) math

Find an expression for v(t) and x(t)

math . \qquad a=12 \, \cos(2t) math . math \\ . \qquad v= \displaystyle{ \int 12 \, \cos(2t) \; dt } \\. \\ . \qquad v=6 \, \sin(2t)+c math

At t = 0, v = 0, so c = 0 math . \qquad v=6 \, \sin(2t) math

math \\ . \qquad x= \displaystyle{ \int 6 \,\sin(2t) \; dt} \\. \\ . \qquad x=-3 \, \cos(2t)+c math

At t = 0, x = 2, so c = 5 math . \qquad x=-3 \, \cos(2t)+5 math

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