7. A particle's velocity is described by v(t) = (et sin(t), et, et cos(t)) for t = [0, 5]. (a) If the particle's initial position is (-½, 1, 1), describe its position r(t) and acceleration a(t) as functions of t. Hint: (e¹ (sint + cost)) = 2e¹ cost and (e¹ (sin t – cos t)) = 2e¹ sint. (b) Verify that the particle's path lies on a circular cone of the form y² = a²(x² + z²) for some a (which?) and sketch the path. (c) What is the length of the particle's path from t = 0 tot = 5?
7. A particle's velocity is described by v(t) = (et sin(t), et, et cos(t)) for t = [0, 5]. (a) If the particle's initial position is (-½, 1, 1), describe its position r(t) and acceleration a(t) as functions of t. Hint: (e¹ (sint + cost)) = 2e¹ cost and (e¹ (sin t – cos t)) = 2e¹ sint. (b) Verify that the particle's path lies on a circular cone of the form y² = a²(x² + z²) for some a (which?) and sketch the path. (c) What is the length of the particle's path from t = 0 tot = 5?
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![7. A particle's velocity is described by v(t) = (et sin(t), et, et cos(t)) for t = [0, 5].
(a) If the particle's initial position is (-½, 1, 1), describe its position r(t) and acceleration a(t) as
functions of t. Hint: (et (sint + cost)) = 2e cost and (et (sint - cost)) = 2e¹ sint.
(b) Verify that the particle's path lies on a circular cone of the form y² = a²(x² + z²) for some a
(which?) and sketch the path.
(c) What is the length of the particle's path from t = 0 tot = 5?
(d) Parameterize the particle's position in terms of arc length.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90076510-2334-4529-9529-9146d67f970e%2F95b6197a-e1c7-417e-a69c-fa1e3bfcd669%2Flx3v5g8_processed.png&w=3840&q=75)
Transcribed Image Text:7. A particle's velocity is described by v(t) = (et sin(t), et, et cos(t)) for t = [0, 5].
(a) If the particle's initial position is (-½, 1, 1), describe its position r(t) and acceleration a(t) as
functions of t. Hint: (et (sint + cost)) = 2e cost and (et (sint - cost)) = 2e¹ sint.
(b) Verify that the particle's path lies on a circular cone of the form y² = a²(x² + z²) for some a
(which?) and sketch the path.
(c) What is the length of the particle's path from t = 0 tot = 5?
(d) Parameterize the particle's position in terms of arc length.
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