Tuesday, February 07, 2006

Slightly More Complicated Motion

How do we describe motion with non-constant acceleration?

We can still get the instantaneous velocity by taking Deltax/Deltat and taking the limit as Deltat --> 0. This is the slope of the tangent to the curve at the point in question. At point a, the slope is positive: the particle has a positive velocity. At b, the slope is still positive, but shallower: the particle has a smaller positive velocity – it is still going in the same direction, only slower.

At c, the particle is stationary ( albeit only for an instant). We know this because the tangent to the curve here is horizontal. At d, the slope of the tangent is negative, so the particle is moving in the negative direction.

If you are familiar with calculus, you will know that the slope of the tangent is the same as the derivative of the curve: v(t) = dx(t)/dt, which is easy to calculate if you can write x(t) out as a function.


    i. x(t) = t --> v(t) = 1

    ii. x(t) = 3t2 --> v(t) = 6t

    iii. x(t) = e2x --> v(t) = 2e2x

    iv. x(t) = cos(x) --> v(t) = -sin(x)

Constant Velocity: straight line motion: x(t)=x0+v0t. At all times, the instantaneous velocity is v0.

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