A particle of mass m is moving along the line y-b with constant acceleration a. The areal velocity of the position vector of the particle at time t is `(u=0)`

Options

A. constant
B. `(abt)/(2)`
C. `(abt)/(2m)`
D. `(a^(2)bt)/(2m)`

Answer

To solve the problem step by step, we need to find the areal velocity of a particle moving along the line \( y = b \) with constant acceleration \( a \).

### Step 1: Understand Areal Velocity
Areal velocity is defined as the area swept out per unit time by the position vector of the particle. Mathematically, it can be expressed as:

\[ \text{Areal Velocity} = \frac{A}{t} = \frac{L}{2m} \]

where \( A \) is the area swept out, \( t \) is the time, \( L \) is the angular momentum, and \( m \) is the mass of the particle.

### Step 2: Express Angular Momentum
The angular momentum \( L \) of a particle can be expressed in terms of its position vector \( \vec{r} \) and momentum \( \vec{p} \):

\[ L = \vec{r} \times \vec{p} \]

where \( \vec{p} = m\vec{v} \) is the momentum of the particle.

### Step 3: Substitute for Areal Velocity
Substituting the expression for angular momentum into the equation for areal velocity gives:

\[ \frac{A}{t} = \frac{L}{2m} = \frac{\vec{r} \times (m\vec{v})}{2m} = \frac{\vec{r} \times \vec{v}}{2} \]

### Step 4: Find the Velocity
Given that the particle starts from rest (\( u = 0 \)), we can find the velocity \( v \) at time \( t \) using the equation of motion:

\[ v = u + at = 0 + at = at \]

### Step 5: Determine the Position Vector
Since the particle is moving along the line \( y = b \), the position vector \( \vec{r} \) can be represented as:

\[ \vec{r} = (x, b) \]

### Step 6: Calculate the Areal Velocity
The perpendicular distance \( r \) from the line of motion to the origin is \( b \). Thus, substituting \( v = at \) and \( r = b \) into the areal velocity formula:

\[ \frac{A}{t} = \frac{(b)(at)}{2} = \frac{abt}{2} \]

### Step 7: Final Expression for Areal Velocity
Thus, the areal velocity becomes:

\[ \text{Areal Velocity} = \frac{ab}{2} \]

### Conclusion
The areal velocity of the position vector of the particle at time \( t \) is:

\[ \frac{ab}{2} \]