a) The velocity of the particle is ⃑'() = 〈2, 0, -16〉. b) The acceleration of the particle is ⃑''() = 〈0, 0, 0〉. c) The particle will never come to a complete stop. d) The speed of the particle is at a minimum and equal to 8√5.
a) The velocity of a particle is the derivative of its position function concerning time. In this case, the given position function is ⃑() = 〈2, 5, 2 − 16〉. Therefore, the velocity function ⃑'() can be found by taking the derivative of each component: ⃑'() = 〈2, 0, -16〉.
b) The acceleration of a particle is the derivative of its velocity function concerning time. Using the velocity function ⃑'() = 〈2, 0, -16〉, the acceleration function ⃑''() can be found by taking the derivative of each component: ⃑''() = 〈0, 0, 0〉. Since all components of the acceleration function are zero, the particle has no acceleration.
c) Since the particle has no acceleration and its velocity is non-zero, the particle will never come to a complete stop.
d) The speed of the particle at a minimum will occur when the magnitude of the velocity vector is at a minimum. Since the velocity vector ⃑'(t) = 〈2, 0, -16〉 has a constant magnitude of √(2² + 0² + (-16)²) = √320 = 8√5, the speed of the particle is at a minimum and equal to 8√5.