Final answer:
To find the velocity of a particle, differentiate the position function to get the velocity function. Differentiate the velocity function to get the acceleration function. The speed of the particle is the magnitude of the velocity vector.
Step-by-step explanation:
To find the velocity of a particle, we differentiate the position function with respect to time. Therefore, the velocity function v(t) is obtained by taking the derivative of the position function:
v(t) = -8 sin(t)î + 6ĵ + 8 cos(t)Ķ
To find the acceleration, we differentiate the velocity function with respect to time. Therefore, the acceleration function a(t) is obtained by taking the derivative of the velocity function:
a(t) = -8 cos(t)î + 8 sin(t)ĵ - 8 sin(t)Ķ
The speed of the particle is the magnitude of the velocity vector:
speed = √((-8 sin(t))^2 + (6)^2 + (8 cos(t))^2)
Therefore, the velocity, acceleration, and speed of the particle can be determined using the given position function.