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

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

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

© BrainMass Inc. brainmass.com September 29, 2022, 1:27 pm ad1c9bdddf
https://brainmass.com/math/vector-calculus/velocity-speed-acceleration-vector-valued-function-69792

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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 = Π/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= Π/4, we substitute the value in (1) to get :

v(t=Π/4) = -sinΠ/4 i + cosΠ/4 j - 32x Π/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 = √sin2t + cos2t + (32t)2 = √1+(32t)2

(Because sin2t + cos2t = 1)
__________
At t = Π/4, s = √1+(32xΠ/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= Π/4, a(t=Π/4) = -cosΠ/4 i - sinΠ/4 j - 32 k = -0.707 i - 0.707 j - 32 k m/s2

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

© BrainMass Inc. brainmass.com September 29, 2022, 1:27 pm ad1c9bdddf>
https://brainmass.com/math/vector-calculus/velocity-speed-acceleration-vector-valued-function-69792