Suppose r? (t)=cos(?t)i+sin(?t)j+4tk represents the position of a particle on a

Question

Suppose r? (t)=cos(?t)i+sin(?t)j+4tk represents the position of a particle on a helix, where z is the height of the particle. (a) What is t when the particle has height 16? t= 4 (b) What is the velocity of the particle when its height is 16? v? = <(-sin(4pi))pi, (cos(4pi))pi, 4> (c) When the particle has height 16, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line. L(t)=

Explanation / Answer

Solution:- We have r(t) = cos(pi t) i + sin(pit) j + 4t k

(a) The variable z represents the height.
So, we put 4t = 16 , which gives us t = 4.

(b) Also Differentiating r(t) with to t we get v(t) = r'(t) = <-pi sin(pi t), pi cos(pi t), 4>.

at t=4 the velocity of the partice is  v(4) = <0, -pi 4>.

(c) Since r(4) = <1, 0, 16>, and r'(4) = <0, -pi, 4>,

Therefore the tangent line has equation
L(t) = <1, 0, 16> + t<0, -?, 4> = <1, -pi t, 4t + 16>.

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