Two cars A and B are moving on parallel roads in the same direction. Car A moves with constant velocity ` 30m//s` and car B moves with constant acceleration `2.5 m//s^(2) ` . At , car A is ahead of car B and car B is moving with ` 12 m//s` velocity . The distance travelled by car B until it overtakes car A is

Explanation

To solve the problem step by step, we will analyze the motion of both cars and set up equations based on their respective motions.

### Step 1: Define the variables and initial conditions
- Let the distance travelled by car B until it overtakes car A be \( x \).
- Car A moves with a constant velocity \( v_A = 30 \, \text{m/s} \).
- Car B starts with an initial velocity \( u_B = 12 \, \text{m/s} \) and has a constant acceleration \( a_B = 2.5 \, \text{m/s}^2 \).

### Step 2: Write the equation for the distance travelled by car A
Since car A moves with constant velocity, the distance travelled by car A in time \( t \) is given by:
\[ x = v_A \cdot t = 30t \]

### Step 3: Write the equation for the distance travelled by car B
Car B, which is accelerating, uses the equation of motion:
\[ x = u_B \cdot t + \frac{1}{2} a_B \cdot t^2 \]
Substituting the values for \( u_B \) and \( a_B \):
\[ x = 12t + \frac{1}{2} \cdot 2.5 \cdot t^2 = 12t + 1.25t^2 \]

### Step 4: Set the equations equal to each other
Since both cars travel the same distance \( x \) when car B overtakes car A, we can set the two equations equal:
\[ 30t = 12t + 1.25t^2 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 30t - 12t - 1.25t^2 = 0 \]
\[ 18t - 1.25t^2 = 0 \]

### Step 6: Factor out \( t \)
Factoring out \( t \):
\[ t(18 - 1.25t) = 0 \]
This gives us two solutions: \( t = 0 \) (the starting point) or \( 18 - 1.25t = 0 \).

### Step 7: Solve for \( t \)
Solving \( 18 - 1.25t = 0 \):
\[ 1.25t = 18 \implies t = \frac{18}{1.25} = 14.4 \, \text{s} \]

### Step 8: Substitute \( t \) back to find \( x \)
Now substitute \( t = 14.4 \) back into the equation for \( x \):
\[ x = 30t = 30 \cdot 14.4 = 432 \, \text{m} \]

### Final Answer
The distance travelled by car B until it overtakes car A is \( \boxed{432 \, \text{m}} \).