Path of a particle moving in `x-y` plane is `y=3x+4`. At some instant suppose `x`- component of velocity is `1m//s` and it is increasing at a constant rate of `1m//s^(2)`. Then at this instant.

Options

A. (a)speed of particle is `sqrt(10)m//s`
B. (b)acceleration of particle is `sqrt(10)m//s`
C. (c)velocity time graph is parabola
D. (d)acceleration time graph is parabola

Answer

To solve the problem step by step, we will analyze the motion of a particle in the x-y plane given the equation of its path and the conditions of its velocity and acceleration.

### Step 1: Understand the path of the particle
The path of the particle is given by the equation:
\[ y = 3x + 4 \]
This is a linear equation, which represents a straight line in the x-y plane.

### Step 2: Determine the slope of the path
The slope of the line can be determined from the equation. The slope (m) is 3, which means for every unit increase in x, y increases by 3 units.

### Step 3: Find the angle of inclination
The angle \( \theta \) that the line makes with the x-axis can be found using the tangent of the angle:
\[ \tan(\theta) = \text{slope} = 3 \]
Thus,
\[ \theta = \tan^{-1}(3) \]

### Step 4: Analyze the velocity components
At the given instant, the x-component of velocity \( v_x \) is:
\[ v_x = 1 \, \text{m/s} \]
It is also given that \( v_x \) is increasing at a constant rate of:
\[ a_x = 1 \, \text{m/s}^2 \]

### Step 5: Relate the y-component of velocity
Since the particle is moving along the path \( y = 3x + 4 \), we can find the y-component of the velocity \( v_y \) using the relationship between the components of velocity and the slope of the path:
\[ \frac{v_y}{v_x} = \text{slope} = 3 \]
Thus,
\[ v_y = 3v_x = 3 \times 1 = 3 \, \text{m/s} \]

### Step 6: Find the total velocity
The total velocity \( v \) can be calculated using the Pythagorean theorem:
\[ v = \sqrt{v_x^2 + v_y^2} = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \, \text{m/s} \]

### Step 7: Analyze the acceleration components
The acceleration in the y-direction \( a_y \) can be found using the same slope relationship:
\[ \frac{a_y}{a_x} = \text{slope} = 3 \]
Thus,
\[ a_y = 3a_x = 3 \times 1 = 3 \, \text{m/s}^2 \]

### Step 8: Summary of results
At the instant described:
- The x-component of velocity \( v_x = 1 \, \text{m/s} \)
- The y-component of velocity \( v_y = 3 \, \text{m/s} \)
- The total velocity \( v = \sqrt{10} \, \text{m/s} \)
- The x-component of acceleration \( a_x = 1 \, \text{m/s}^2 \)
- The y-component of acceleration \( a_y = 3 \, \text{m/s}^2 \)