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
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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© BrainMass Inc. brainmass.com September 29, 2022, 1:27 pm ad1c9bdddf>