Question

Describe the motion of a particle with position (x, y) as t varies in the given interval. (for each answer, enter an ordered pair of the form x, y.) x = 3 sin(t), y = 5 2 cos(t), /2 ≤ t ≤ 2?

Answer 1

Describing the motion of a particle involves using position functions to determine how variables like position, velocity, and acceleration change over time. Example problems show different motion scenarios like crossing the origin, finding displacement, and circular motion with centripetal acceleration.

Step-by-step explanation:

Motion of a Particle

The question of describing the motion of a particle with a given position function involves understanding how position, velocity, and acceleration change with time. Particle motion can be linear, along a plane, or in three dimensions, and is always defined by its position as a function of time, typically represented as x(t), y(t), and z(t) for three-dimensional motion.

Example Problems

• To find when a particle crosses the origin with position x(t) = 4.0 - 2.0t, set the position equation equal to zero and solve for t.
• To measure displacement over a time interval, evaluate the position function at the beginning and end of the interval and calculate the difference.
• For circular motion, velocity and acceleration can be determined by differentiating the position function with respect to time, and the centripetal acceleration will always point towards the center of the circle.

Using these principles, we can analyze various motion scenarios, determine velocities and accelerations, and describe the motion paths of particles under different conditions. Whether it's linear motion, circular motion, or wave patterns, physics provides the tools to describe the movement of particles comprehensively.