Final answer:
The velocity v(t) and speed ∥v(t)∥ of the particle can be found by differentiating the position vector and calculating its magnitude.
Step-by-step explanation:
To find the velocity v(t) and speed ∥v(t)∥ of the particle, we need to differentiate the given position vector with respect to time.
Given position vector: x = 8t^3 - 36t^2, y = 3t^2 - 18t + 27, z = 2
Velocity vector: v(t) = (dx/dt)i + (dy/dt)j + (dz/dt)k
By differentiating each component of the position vector, we get:
dx/dt = 24t^2 - 72t
dy/dt = 6t - 18
dz/dt = 0
Therefore, the velocity vector is given by:
v(t) = (24t^2 - 72t)i + (6t - 18)j
To find the speed, we calculate the magnitude of the velocity vector:
∥v(t)∥ = √((24t^2 - 72t)^2 + (6t - 18)^2)
Learn more about Velocity and Speed