** Please see the attached file for the complete problem description **
Velocity and acceleration: consider a velocity vector given in standard unit vector form in rectangular Cartesian coordinates as ____ (please see the attached file).
a. What is the acceleration at 2 seconds?
b. What is the displacement at 5 seconds?
c. What is the angle between the displacement and the acceleration at t=2.0s?
d. What is the velocity of the vector at 2 seconds?
e. What is the magnitude of the displacement at 5 seconds?
** Please see the attachment for a Word formatted solution **
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Velocity vector v(t) = 3t4 i + 7t5/2 j + 5e2t k .....(1)
Displacement vector s(t) = ∫(3t4 i + 7t5/2 j + 5e2t k) dt
s(t) = 3∫(t4 dt ) i + 7∫(t5/2 dt) j + 5∫e2t dt) k
s(t) = 3(t5/5) i + 7 x (2/7) t7/2 j + 5 (e2t/2) k
s(t) = 3t5/5 i + 2t7/2 j + 5e2t/2 k ......(2)
Acceleration vector a(t) = dv/dt = d/dt(3t4 i + 7t5/2 j + 5e2t k)
a(t) = 3x4t3 i + 7x5/2xt3/2 j + 5x2xe2t k
a(t) = 12t3 i + 35/2xt3/2 j + 10e2t k ...
This solution involves time varying displacement, velocity and acceleration vectors. For a given velocity vector, expressions for the displacement and acceleration vectors have been derived and evaluated for different time instants.