Question

The motion of a particle is given by \(X = a \cos t\), \(Y = a \sin t\) and \(Z = t\). The trajectory traced by the particle as a function of time is:

• A. Helix
• B. Circular
• C. Elliptical
• D. Straight line

Hint:

Helical trajectories are common in physics, representing combined rotational and translational motion, such as in a spring or screw.

Answer 1

The Correct Option is A

Step 1: Analyze the motion in the \( XY \) plane.
The parametric equations \( X = a \cos t \) and \( Y = a \sin t \) describe a circle in the \( XY \) plane.

Step 2: Consider the motion along \( Z \).
The \( Z \) coordinate increases linearly with time \( t \), indicating a vertical motion component.

Step 3: Combine the motions.
Combining the circular motion in the \( XY \) plane with the linear increase in \( Z \) gives a helical trajectory.

Answer 2

To solve the problem, we need to analyze the given motion equations and identify the trajectory of the particle.

1. Understanding the Motion Equations:
The motion of the particle is given by the following equations:
\( X = a \cos t \)
\( Y = a \sin t \)
\( Z = t \)
These equations represent the motion in the three-dimensional space with:
\( X \) and \( Y \) describing a circular motion in the \( XY \)-plane, and
\( Z \) representing linear motion along the \( Z \)-axis. This describes a helix, where the particle moves in a spiral path, alternating between circular motion in the \( XY \)-plane and linear motion along the \( Z \)-axis.

2. Conclusion:
Based on the analysis of the motion equations, the trajectory of the particle is a helix.

Final Answer:
The correct option is (A) Helix.