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Consider the motion of a particle de...

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

A

Helix

B

Circular

C

Elliptical

D

Straight line

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the motion of the particle described by the equations: 1. \( x = a \cos t \) 2. \( y = a \sin t \) 3. \( z = t \) ### Step 1: Understand the equations The equations represent the coordinates of the particle in three-dimensional space as functions of time \( t \). Here, \( a \) is a constant that determines the radius of the circular motion in the \( xy \)-plane. ### Step 2: Analyze the \( xy \)-plane motion To understand the motion in the \( xy \)-plane, we can eliminate the parameter \( t \) by using the trigonometric identity: \[ x^2 + y^2 = (a \cos t)^2 + (a \sin t)^2 \] ### Step 3: Simplify the equation By simplifying the equation, we have: \[ x^2 + y^2 = a^2 (\cos^2 t + \sin^2 t) \] Using the identity \( \cos^2 t + \sin^2 t = 1 \): \[ x^2 + y^2 = a^2 \] This equation represents a circle of radius \( a \) in the \( xy \)-plane. ### Step 4: Analyze the \( z \)-axis motion The \( z \) coordinate is given by: \[ z = t \] This indicates that as time \( t \) increases, the \( z \) coordinate increases linearly. Thus, the particle moves upward along the \( z \)-axis as it moves in a circular path in the \( xy \)-plane. ### Step 5: Combine the motions Since the particle moves in a circular path in the \( xy \)-plane while simultaneously moving linearly along the \( z \)-axis, the overall trajectory of the particle is a helix. ### Conclusion The trajectory traced by the particle as a function of time is a helix. ### Final Answer The correct option is: **Helix**. ---
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