Question

\(x\) and y co-ordinates of a particle moving in \(\mathrm{x}-\mathrm{y}\) plane at some instant are \(\mathrm{x}=2 \mathrm{t}^{2}\) and \(\mathrm{y}=(3 / 2) \mathrm{t}^{2}\) Calculate y co-ordinate when its \(\mathrm{x}\) coordinate is \(8 \mathrm{~m}\). (A) \(3 \mathrm{~m}\) (B) \(6 \mathrm{~m}\) (C) \(8 \mathrm{~m}\) (D) \(9 \mathrm{~m}\)

Short answer

The y-coordinate of the particle when its x-coordinate is 8 m is \(6 \mathrm{~m}\).

Step-by-step solution

Analyze the given equations for x and y

The equations given for the x and y coordinates of the particle are: \[x = 2t^2\] \[y = \frac{3}{2}t^2\]

Find the relationship between x and y coordinates

We can find the relationship between x and y coordinates by eliminating t from the given equations. To do that, we can express t^2 in terms of x and substitute it into the y equation. Rearranging the x equation: \[t^2 = \frac{x}{2}\] Now substitute this expression into the y equation: \[y = \frac{3}{2} (\frac{x}{2})\]

Simplify the relationship equation

Now, let's simplify the equation we got in Step 2: \[y = \frac{3}{2} \times \frac{x}{2}\] \[y = \frac{3}{4}x\] We have now established a relationship between the x and y coordinates of the particle.

Substitute x = 8 m into the equation and find the corresponding y-coordinate

Now that we have the equation relating x and y, we can substitute x = 8 m into the equation to find the corresponding y-coordinate: \[y = \frac{3}{4}(8)\] \[y = 6\] Therefore, when the x-coordinate of the particle is 8 m, its y-coordinate is 6 m. So, the correct answer is (B) \(6 \mathrm{~m}\).

Key concepts

Coordinate System

In the study of particle motion, understanding the coordinate system is crucial. A coordinate system allows us to define a particle's position in space using numerical values. In this exercise, we're focusing on a two-dimensional (2D) coordinate system, also known as the xy-plane. Here, every position of the particle is described by two coordinates: \(x\) and \(y\).
The \(x\)-coordinate represents the particle's position along the horizontal axis, while the \(y\)-coordinate represents the position along the vertical axis. By analyzing how these coordinates change over time, we can understand the motion of the particle across the plane.
This simple layout aids in visualizing movement and transferring mathematical descriptions into physical interpretations.

Equations of Motion

The equations of motion provide us with a mathematical way to describe how a particle moves in a coordinate system. In this problem, the particle's position is given by two specific equations:

• \(x = 2t^2\)
• \(y = \frac{3}{2}t^2\)

These equations show how the coordinates \(x\) and \(y\) change with respect to time \(t\).
The form \(x = 2t^2\) indicates that the \(x\)-coordinate changes quadratically as time progresses. Similarly, \(y\) depends on \(t^2\) but with a different constant multiplier.
Understanding these equations allows us to predict the future position of the particle and analyze its trajectory on the xy-plane.

Algebraic Manipulation

Algebraic manipulation is key to finding the solution by transforming given equations into useful forms. In this exercise, we were tasked with finding the \(y\)-coordinate when \(x = 8\, m\).
The first step was to express \(t^2\) in terms of \(x\) from the equation \(x = 2t^2\). This gives \(t^2 = \frac{x}{2}\).
Next, substituting \(t^2\) into the \(y\) equation allows us to express \(y\) in terms of \(x\) alone. The new equation becomes \(y = \frac{3}{4}x\).
By simplifying this relationship, we can easily find \(y\) for any given \(x\). This algebraic process is a powerful method to connect different variables and solve problems in particle motion. For \(x = 8\, m\), it led directly to \(y = 6\, m\), providing the solution to our problem.