Question

A particle moves along the plane curve C described by r(t) = i + 12j. Solve the following problems:

a)Find the length of the curve over the interval [0, 2].
b)the curvature of the plane curve at t = 0, 1, 2. c) Describe the curvature as t increases from t = 0 to t = 2.

Answer 1

The questions pertain to kinematics in Physics and involve the computation of velocity, acceleration, and force vectors for a particle moving in a plane. The velocity and acceleration are found by differentiation of position with respect to time, and centripetal force is determined using Newton's second law.

Step-by-step explanation:

The student's question relates to the motion of a particle in a plane, which is a topic in Physics, specifically in mechanics and kinematics. The questions provided seem to be incorrectly stated for the specifics needed; however, I can provide similar answers based on the challenge problems listed:

a) To find the velocity and acceleration vectors as functions of time for a particle moving in a circular path represented by r(t) = (4.0 cos 3t)i + (4.0 sin 3t)j, we would differentiate the position vector with respect to time to find the velocity v(t) and then differentiate the velocity with respect to time to find acceleration a(t). The formulas would be:

• Velocity: v(t) = dr/dt
• Acceleration: a(t) = dv/dt

b) To show that the acceleration vector points towards the center of the circle (centripetal acceleration), we would show that it is proportional to the negative of the position vector, which indicates it points towards the center.

c) The centripetal force vector as a function of time can be found using Newton's second law, F = ma, where m is the mass of the particle and a is the centripetal acceleration.