# Velocity, speed and acceleration of a vector-valued function

Find the velocity, speed and acceleration of the vector-valued function:

Vector r = cost i + sint j - 16t^2 k at t=pi/4

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SOLUTION

Given vector : r = cost i + sint j - 16t2 k

We assume the unit of displacement r as meters and time t as sec.

To determine velocity, speed and acceleration at t = Î /4.

a) Velocity is defined as the rate of charge of displacement vector r.

Velocity v = dr/dt = -sint i + cost j - 32t k m/s ............(1)

To determine value of v at t= Î /4, we substitute the value in (1) to get :

v(t=Î /4) = -sinÎ /4 i + cosÎ /4 j - 32x Î /4 k = -0.707 i + 0.707 j -25.13 k m/s

b) Speed is the magnitude of velocity vector.

_________________ _______

Speed s = magnitude of vector v = âˆšsin2t + cos2t + (32t)2 = âˆš1+(32t)2

(Because sin2t + cos2t = 1)

__________

At t = Î /4, s = âˆš1+(32xÎ /4)2 = 25.15 m/s

c) Acceleration is defined as the rate of change of velocity.

Acceleration a = dv/dt = -cost i - sint j - 32 k m/s2..........(2)

At t= Î /4, a(t=Î /4) = -cosÎ /4 i - sinÎ /4 j - 32 k = -0.707 i - 0.707 j - 32 k m/s2

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