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(A) Given the parametric equations of motion:$x = a \cos t$$y = a \sin t$$z = t$First, consider the projection of the motion on the $xy$-plane:$x^2 +

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helix

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(A) Given the parametric equations of motion:$x = a \cos t$$y = a \sin t$$z = t$First, consider the projection of the motion on the $xy$-plane:$x^2 + y^2 = (a \cos t)^2 + (a \sin t)^2 = a^2(\cos^2 t + \sin^2 t) = a^2$This represents a circle of radius $a$ in the $xy$-plane.Simultaneously, the particle moves along the $z$-axis with a constant velocity since $z = t$, which implies $\frac{dz}{dt} = 1$.Since the particle moves in a circular path in the $xy$-plane while simultaneously advancing linearly along the $z$-axis, the resulting trajectory is a helix.

Consider the motion of a particle described by $x = a \cos t$, $y = a \sin t$, and $z = t$. The trajectory traced by the particle as a function of time is

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