The motion of a particle in the x-y plane can be described by the equations x = asin(ωt) and y = acos(ωt), where a is the amplitude of oscillation and ω is the angular frequency.


The equations x = asin(ωt) and y = acos(ωt) represent the position of the particle in the x and y directions respectively at a given time t. The sinusoidal functions, sin(ωt) and cos(ωt), represent oscillatory motion.

x = asin(ωt):
- The value of x varies with time t and the angular frequency ω.
- The amplitude a determines the maximum displacement of the particle in the x direction.
- The angular frequency ω determines the speed at which the particle oscillates in the x direction. A higher ω corresponds to a faster motion.

y = acos(ωt):
- The value of y also varies with time t and the angular frequency ω.
- The amplitude a determines the maximum displacement of the particle in the y direction.
- The angular frequency ω determines the speed at which the particle oscillates in the y direction. A higher ω corresponds to a faster motion.


When the particle's motion is plotted on a graph with x and y axes, it follows a straight line. This can be observed by combining the equations for x and y:

x = asin(ωt)
y = acos(ωt)

By squaring both equations and adding them, we get:

x^2 + y^2 = a^2(sin^2(ωt) + cos^2(ωt))

Using the trigonometric identity sin^2(θ) + cos^2(θ) = 1, the equation simplifies to:

x^2 + y^2 = a^2

This equation represents a circle with radius a centered at the origin (0, 0) in the x-y plane. Therefore, the particle's motion traces out a circular path.

However, when we plot the particle's motion by varying time t, we observe that the particle moves along a straight line rather than a complete circle. This happens because the x and y components of the particle's motion are out of phase with each other. As a result, the particle's position traces a straight line segment instead of a complete circle.


In conclusion, a particle moving in the x-y plane according to the equations x = asin(ωt) and y = acos(ωt) follows a straight line segment on a graph. The motion can be visualized as a particle moving along a circular path but only tracing a straight line segment due to the out of phase relationship between the x and y components of its motion.