A particle is moving with velocity ` vecv = k( y hat(i) + x hat(j)) `, where `k` is a constant . The genergal equation for its path is

Options

A. `y=x^(2)+"constant"`
B. `y^(2)=x+"constant"`
C. `xy="constant"`
D. `y^(2)=x^(2)+"constant"`

Answer

To find the general equation for the path of a particle moving with the given velocity vector \( \vec{v} = k(\hat{y} + \hat{x}) \), where \( k \) is a constant, we can follow these steps:

### Step 1: Identify the components of the velocity vector
The velocity vector can be expressed in terms of its components:
\[ \vec{v} = k y \hat{i} + k x \hat{j} \]
This means:
- The x-component of the velocity \( v_x = k y \)
- The y-component of the velocity \( v_y = k x \)

### Step 2: Write the differential forms of the components
From the definitions of velocity, we can write:
\[ v_x = \frac{dx}{dt} = k y \quad \text{(1)} \]
\[ v_y = \frac{dy}{dt} = k x \quad \text{(2)} \]

### Step 3: Form the ratio of the differentials
To eliminate \( dt \), we can form the ratio of the two equations:
\[ \frac{dy}{dt} \div \frac{dx}{dt} = \frac{k x}{k y} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{x}{y} \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ y \, dy = x \, dx \]

### Step 5: Integrate both sides
Now, we integrate both sides:
\[ \int y \, dy = \int x \, dx \]
This results in:
\[ \frac{y^2}{2} = \frac{x^2}{2} + C \]
where \( C \) is the constant of integration.

### Step 6: Simplify the equation
Multiplying through by 2 to eliminate the fractions gives:
\[ y^2 = x^2 + 2C \]
We can denote \( 2C \) as a new constant \( k \):
\[ y^2 = x^2 + k \]

### Final Result
Thus, the general equation for the path of the particle is:
\[ y^2 = x^2 + C \]
where \( C \) is a constant.

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