The position vector r(t) = <<lnt, 1/t^2, t^4>> describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t = sqrt3

sagnuhh 2020-12-29 Answered
The position vector r(t)=lnt,1t2,t4 describes the path of an object moving in space.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of t=3
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Expert Answer

Caren
Answered 2020-12-30 Author has 96 answers
a) Velocity vector is the derivative of position vector. So to find velocity vector, differentiate position vector with respect to t. We get v(t)=lnt,1t2,t4
Now speed is the magnitude of velocity vector. That is,
Speed = ||v||
=(1t)2+(2t3)2+(4t3)2
=1t2+4t6+16t6
=t4+4+16t12t6
=16t12+t4+4t3
Acceleration vector is the derivative of the velocity vector.So to find acceleration vector, differentiate velocity vector with respect to t. We get,
a(t)=1t2,2(3t4),4(3t2)=1t2,6t4,12t2
b) Put t=3 in the velocity vector, we get
v(3)=13,2(3)3,4(3)3=13,233,123
Put t=3 ⎯ in the acceleration vector, we get
a(3)=1(3)2,63,12(3)2=13,69,12(3)=13,23,36
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