Final answer:
The velocity of a particle with the position function r(t) = t i + t² j + 2k is v(t) = i + 2t j. Its acceleration is a constant a(t) = 2j, and the speed is a function of time, given by the magnitude of the velocity vector.
Step-by-step explanation:
To find the velocity, acceleration, and speed of a particle with the given position function r(t) = t i + t² j + 2k, we need to differentiate the position vector with respect to time.
Velocity
The velocity vector v(t) is the first derivative of the position vector r(t) with respect to time. Thus:
v(t) = dr(t)/dt = i + 2t j + 0k
This gives us the velocity vector in terms of i, j, and k unit vectors.
Acceleration
The acceleration vector a(t) is the derivative of the velocity vector with respect to time. Consequently, it is:
a(t) = dv(t)/dt = 0i + 2j + 0k
This represents the particle's acceleration at any given time t.
Speed
The speed of the particle is the magnitude of the velocity vector and is given by:
Speed = |v(t)| = √(1² + (2t)²)
Speed is a scalar quantity representing how fast the particle is moving at time t.