91Ó°ÊÓ

Understanding the velocity vector components of a particle moving in two dimensions is crucial to describing how it speeds up, slows down, or changes direction. In our scenario, each position of the particle is given in terms of variables that define both the x and y coordinates. For a position described as \( x = A \cos(bf) \) and \( y = Bt \), the respective velocity components are determined by how these position coordinates change with time.

To determine the velocity vector components, we need to differentiate the position equations with respect to time \( t \):

• The x-component of the velocity \( v_x \) is found by differentiating \( x = A \cos(bf) \), resulting in \( v_x = -Abf \sin(bf) \).
• Similarly, the y-component \( v_y \) is obtained by differentiating \( y = Bt \), giving \( v_y = B \).

Together, these components \( v_x = -Abf \sin(bf) \) and \( v_y = B \) form the complete velocity vector in the plane.

Just like velocity, acceleration can also be broken down into components that align with the x and y axes, showing how they change at any instant. The acceleration components tell us how quickly the velocity itself is changing over time. This is done by differentiating the velocity components we've just calculated.

Let's handle each component:

• The x-component of acceleration \( a_x \) is found by differentiating \( v_x = -Abf \sin(bf) \) with respect to time, yielding \( a_x = -Ab^2f^2 \cos(bf) \). This indicates how the velocity in the x-direction is changing.
• The y-component of acceleration \( a_y \) comes from differentiating \( v_y = B \), which results in \( a_y = 0 \). It means the velocity in the y-direction is constant and does not change over time.

Thus, the acceleration vector consists of \( a_x = -Ab^2f^2 \cos(bf) \) and \( a_y = 0 \). These components reflect the particle's dynamics at each moment.

The instantaneous speed of a particle is a measure of how fast it is moving at a specific point in time, regardless of its direction. It is the magnitude of the velocity vector in two-dimensional motion and provides a scalar value that tells us the particle's rate of motion at that instant.

To calculate the instantaneous speed, we use the formula for the magnitude of the velocity vector:
\[v = \sqrt{v_x^2 + v_y^2}\]
For the components we've determined, this calculation becomes:
\[v = \sqrt{(-Abf \sin(bf))^2 + B^2} = \sqrt{A^2b^2f^2 \sin^2(bf) + B^2}\]
By solving this expression, you obtain a numeric value representing the speed at any given moment. This result indicates how quickly the particle is moving through the plane, combining both x and y velocity influences into a single figure of merit.