A car starts from rest and moves with constant acceleration. In first t seconds it covers distance x. Then distance covered by it in next t seconds will be:-

Options

A. x
B. 2 x
C. 3 x
D. 4 x

Answer

To solve the problem, we need to analyze the motion of the car under constant acceleration. Let's break it down step by step.

### Step 1: Understand the motion of the car
The car starts from rest, which means its initial velocity \( u = 0 \). It moves with constant acceleration \( a \).

### Step 2: Use the formula for distance covered under constant acceleration
The formula for the distance \( s \) covered by an object starting from rest under constant acceleration is given by:
\[ s = ut + \frac{1}{2} a t^2 \]
Since the initial velocity \( u = 0 \), the formula simplifies to:
\[ s = \frac{1}{2} a t^2 \]

### Step 3: Distance covered in the first \( t \) seconds
According to the problem, the distance covered in the first \( t \) seconds is \( x \). Therefore, we can write:
\[ x = \frac{1}{2} a t^2 \]

### Step 4: Find the distance covered in the next \( t \) seconds
To find the distance covered in the next \( t \) seconds, we need to find the total distance covered in the first \( 2t \) seconds and then subtract the distance covered in the first \( t \) seconds.

The total distance covered in \( 2t \) seconds is:
\[ s_{2t} = \frac{1}{2} a (2t)^2 = \frac{1}{2} a (4t^2) = 2 a t^2 \]

Now, the distance covered in the next \( t \) seconds (from \( t \) to \( 2t \)) is:
\[ \text{Distance in next } t \text{ seconds} = s_{2t} - s_{t} = 2 a t^2 - \frac{1}{2} a t^2 \]
\[ = 2 a t^2 - \frac{1}{2} a t^2 = \frac{4}{2} a t^2 - \frac{1}{2} a t^2 = \frac{3}{2} a t^2 \]

### Step 5: Relate \( a \) to \( x \)
From the equation \( x = \frac{1}{2} a t^2 \), we can express \( a \) in terms of \( x \):
\[ a = \frac{2x}{t^2} \]

### Step 6: Substitute \( a \) back into the distance formula
Now substituting \( a \) into the distance covered in the next \( t \) seconds:
\[ \text{Distance in next } t \text{ seconds} = \frac{3}{2} \left(\frac{2x}{t^2}\right) t^2 = 3x \]

### Conclusion
Thus, the distance covered by the car in the next \( t \) seconds is \( 3x \).

### Final Answer
The distance covered by the car in the next \( t \) seconds is \( 3x \).

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