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 follow these instructions:

### Step 1: Understand the initial conditions
- Car A moves with a constant velocity of \( v_A = 30 \, \text{m/s} \).
- Car B starts with an initial velocity of \( u_B = 12 \, \text{m/s} \) and has a constant acceleration of \( a_B = 2.5 \, \text{m/s}^2 \).
- At \( t = 0 \), car A is ahead of car B.

### Step 2: Set up the equations for distance traveled
- The distance traveled by car A after time \( t \) is given by:
\[ d_A = v_A \cdot t = 30t \]
- The distance traveled by car B after time \( t \) is given by:
\[ d_B = u_B \cdot t + \frac{1}{2} a_B \cdot t^2 = 12t + \frac{1}{2} \cdot 2.5 \cdot t^2 = 12t + 1.25t^2 \]

### Step 3: Set the distances equal for overtaking
Since car B overtakes car A when they have traveled the same distance, we set \( d_A = d_B \):
\[ 30t = 12t + 1.25t^2 \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ 30t - 12t = 1.25t^2 \]
\[ 18t = 1.25t^2 \]

### Step 5: Simplify the equation
Dividing both sides by \( t \) (assuming \( t \neq 0 \)):
\[ 18 = 1.25t \]

### Step 6: Solve for \( t \)
Now, solve for \( t \):
\[ t = \frac{18}{1.25} = \frac{18 \times 4}{5} = \frac{72}{5} = 14.4 \, \text{s} \]

### Step 7: Calculate the distance traveled by car B
Now, we can find the distance traveled by car B when it overtakes car A:
\[ d_B = u_B \cdot t + \frac{1}{2} a_B \cdot t^2 \]
Substituting the values:
\[ d_B = 12 \cdot 14.4 + \frac{1}{2} \cdot 2.5 \cdot (14.4)^2 \]
Calculating \( \frac{1}{2} \cdot 2.5 \cdot (14.4)^2 \):
\[ = 1.25 \cdot 207.36 = 259.2 \]
So,
\[ d_B = 12 \cdot 14.4 + 259.2 = 172.8 + 259.2 = 432 \, \text{meters} \]

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

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