A particle leaves the origin at t = 0 with an intial velocity `vec(V)=3V_(0)hat(i)`. It experiences a constant acceleration `vec(a)=-2a_(1)hat(i)-5a_(2)hat(j)`. The time at which the particle reaches its maximum x-coordinate in positive direction is (`V_(0),a_(1)` and `a_(2)` are positive constants)

Options

A. `(3V_(0))/(10a_(2))`
B. `(3V_(0))/(10a_(2))`
C. `(3V_(0))/(2a_(1))`
D. `(3V_(0))/(2a_(1)+5a_(2))`

Explanation

To find the time at which the particle reaches its maximum x-coordinate in the positive direction, we can follow these steps:

### Step 1: Understand the motion of the particle
The particle starts at the origin (0,0) with an initial velocity in the x-direction and experiences a constant acceleration. The initial velocity is given as:
\[ \vec{V} = 3V_0 \hat{i} \]
And the acceleration is:
\[ \vec{a} = -2a_1 \hat{i} - 5a_2 \hat{j} \]

### Step 2: Write the equation for velocity
The velocity of the particle at any time \( t \) can be expressed using the equation:
\[ \vec{V}(t) = \vec{V}_0 + \vec{a} t \]
Substituting the values:
\[ \vec{V}(t) = 3V_0 \hat{i} + (-2a_1 \hat{i} - 5a_2 \hat{j}) t \]
This simplifies to:
\[ \vec{V}(t) = (3V_0 - 2a_1 t) \hat{i} - 5a_2 t \hat{j} \]

### Step 3: Find the condition for maximum x-coordinate
The maximum x-coordinate is reached when the x-component of the velocity becomes zero. Thus, we set the x-component of the velocity to zero:
\[ 3V_0 - 2a_1 t = 0 \]

### Step 4: Solve for time \( t \)
Rearranging the equation gives:
\[ 2a_1 t = 3V_0 \]
Dividing both sides by \( 2a_1 \):
\[ t = \frac{3V_0}{2a_1} \]

### Conclusion
The time at which the particle reaches its maximum x-coordinate in the positive direction is:
\[ t = \frac{3V_0}{2a_1} \]