# An object's two dimensional velocity is given by #v(t) = ( t^2 - 2t , cospit - t )#. What is the object's rate and direction of acceleration at #t=2 #?

The rate of acceleration is

the direction is

Acceleration is the velocity derivative.

The acceleration rate is

The path is

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To find the acceleration, differentiate the velocity function with respect to time twice. Then evaluate the resulting expression at ( t = 2 ).

First derivative of velocity: ( v'(t) = (2t - 2, -\sin(pt) \cdot p - 1) )

Second derivative of velocity: ( a(t) = v''(t) = (2, -\cos(pt) \cdot p^2) )

At ( t = 2 ), the acceleration vector is: ( a(2) = (2, -\cos(2p) \cdot p^2) )

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