Final answer:
Velocity and acceleration vectors are found by differentiating the position vector with respect to time. The speed is the magnitude of the velocity vector. For circular motion, acceleration towards the center is centripetal acceleration.
Step-by-step explanation:
To determine velocity and acceleration vectors, we differentiate the position vector ρ(t) with respect to time (t). For the given equations, we do this for each component of the vector separately. From the information provided:
- Velocity vector at a given time can be found by the derivative of the position vector, i.e., dr(t)/dt.
- Acceleration vector is the derivative of the velocity vector with respect to time.
- The speed is the magnitude of the velocity vector.
For the example with r(t) = (4.0 cos 3t)i + (4.0 sin 3t)j, the velocity vector at any time t is the derivative of r(t), and the acceleration vector is the derivative of the velocity. Additionally, the acceleration towards the center of a circle represents centripetal acceleration, and the centripetal force can be found as the mass times the centripetal acceleration vector. For the motion defined by ρ(t) = (50 m/s)tî – (4.9 m/s²)t² ı, the velocity is the first derivative, and acceleration is the second derivative with respect to time.
The functional form of the acceleration can be obtained by differentiating the position vector twice with respect to time. Instantaneous velocity and acceleration at specific times are found by plugging those times into the derivative equations.