Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

검증된 단계별 안내

Recall that the equation of the circle is given by \(x^{2} + y^{2} = a^{2}\). This describes a circle centered at the origin with radius \(a\).

To find parametric equations for the motion of a particle on this circle, use the standard parameterization involving trigonometric functions: \(x = a \cos(t)\) and \(y = a \sin(t)\), where \(t\) is the parameter.

Since the particle starts at the point \((a, 0)\), determine the value of \(t\) at the start. Substitute \(x = a\) and \(y = 0\) into the parametric equations to find the initial parameter value.

To trace the circle once clockwise, consider the direction of motion. The standard parameterization with \(t\) increasing from \(0\) to \(2\pi\) traces the circle counterclockwise. To reverse the direction, modify the parameterization to \(x = a \cos(t)\) and \(y = -a \sin(t)\).

Set the parameter interval for \(t\) to cover one full revolution, typically \(t\) going from \(0\) to \(2\pi\), to ensure the particle traces the entire circle once in the clockwise direction.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations of a Circle

Parametric equations express the coordinates of points on a curve as functions of a parameter, often time. For a circle of radius a centered at the origin, a common parametrization is x = a cos(t), y = a sin(t), where t varies over an interval to trace the circle.

Direction of Motion (Clockwise vs Counterclockwise)

The direction in which a particle moves along a curve depends on how the parameter affects the coordinates. For a circle, using x = a cos(t), y = a sin(t) traces counterclockwise motion, while x = a cos(t), y = -a sin(t) reverses the direction to clockwise.

Parameter Interval and Initial Point

Choosing the correct parameter interval ensures the particle starts at the given point and completes the desired path. For starting at (a, 0), setting t = 0 at this point and letting t vary from 0 to 2π traces the full circle once.

Finding Parametric EquationsFind parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².a. once clockwise.(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)