Problem 3: The r, y, and z coordinates of a particle P as a function of time are

Question

Problem 3: The r, y, and z coordinates of a particle P as a function of time are: r = 2sin(t) (m), y = 6sin(2t) (m), and z = cos(3t) (m) If time is measured in seconds, at time equal to 6 seconds calculate: a. The velocity of the particle in Cartesian coordinates b. The speed of the particle c. The angles that the velocity makes with the x, y , and z axes d. The unit tangent vector e. The acceleration in Cartesian coordinates f. The binormal vector g. The normal vector h. The tangential component of acceleration i. The normal component of acceleration j. The radius of curvature of the path traveled by P

Explanation / Answer

x = 2 sin(t)

y = 6 sin(2t)

z = cos(3t)

(a) velocity = ( dxdt, dy/dt, dz/dt) = ( 2 cos(t) , 12 cos(2t), - 3 sin(3t)

At t = 6 s

velocity = ( 2 cos(6), 12 cos(12), - 3 sin(18) )

(b) speed = sqrtr( 4cos^2(6) + 144cos^2(12) + 9 sin^2(18)) = 10.196

(e) accelerator = ( - 2sin(t) , - 24cos(2t), - 9 cos(3t) )

= ( - 2 sin(6), - 24 cos(12), - 9 cos(18) )

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