Suppose the position vector of a particle in space at time t is given by: r (12t,-312,-8t/2) (a) Find the particle's velocity and acceleration vectors. (b) Find the particle's speed and direction of motion.

Suppose the position vector of a particle in space at time t is given by: r (12t,-312,-8t/2) (a) Find the particle's velocity and acceleration vectors. (b) Find the particle's speed and direction of motion.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.4: The Dot Product

Problem 32E

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Question

4 Suppose the position vector of a particle in space at time t is given by: r =
(12t,-312,-8t/2)
(a) Find the particle's velocity and acceleration vectors.
(b) Find the particle's speed and direction of motion.

Expert Solution

Step 1

$s o l u t i o n : - r (t) = <12 t, - 3 t^{2}, - 8 t^{3 / 2}> (a) t h e p a r t i c l e' s v e l o c i t y = r' (t) = <\frac{d}{d t} (12 t), \frac{d}{d t} (- 3 t^{2}), \frac{d}{d t} (- 8 t^{3 / 2})> r' (t) = <12, - 6 t, - 12 t^{1 / 2}> t h e p a r t i c l e' s a c c e l e r a t i o n = r'' (t) = \frac{d}{d t} (r' (t)) r'' (t) = <\frac{d}{d t} (12), \frac{d}{d t} (- 6 t), \frac{d}{d t} (- 12 t^{1 / 2})> = <0, - 6, - 6 t^{- 1 / 2}>$