A particle moves along the straight line `y=3x+5`. Which coordinate changes at faster rate?

Options

A. PV
B. Pb
C. `a/V^(2)`
D. `(ab)/V^(2)`

Explanation

To solve the problem of which coordinate changes at a faster rate for a particle moving along the straight line given by the equation \(y = 3x + 5\), we will follow these steps:

### Step 1: Understand the relationship between \(x\) and \(y\)
The equation of the line is given as:
\[ y = 3x + 5 \]
This indicates that \(y\) is a linear function of \(x\).

### Step 2: Differentiate both sides with respect to time \(t\)
Assuming both \(x\) and \(y\) are functions of time \(t\), we differentiate the equation with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(3x + 5) \]
Using the chain rule, we differentiate the right-hand side:
\[ \frac{dy}{dt} = 3 \frac{dx}{dt} + 0 \]
The term \(0\) comes from the derivative of the constant \(5\).

### Step 3: Simplify the equation
From the differentiation, we have:
\[ \frac{dy}{dt} = 3 \frac{dx}{dt} \]

### Step 4: Analyze the rates of change
This equation shows that the rate of change of \(y\) with respect to \(t\) is three times the rate of change of \(x\) with respect to \(t\):
\[ \frac{dy}{dt} = 3 \frac{dx}{dt} \]
This means that for every unit change in \(x\), \(y\) changes three units.

### Step 5: Conclusion
Since \(\frac{dy}{dt}\) is greater than \(\frac{dx}{dt}\), we conclude that the \(y\) coordinate changes at a faster rate than the \(x\) coordinate.

### Final Answer:
The \(y\) coordinate changes at a faster rate.

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