A particle start moving from rest state along a straight line under the action of a constant force and travel distance x in first 5 seconds. The distance travelled by it in next five seconds will be

Options

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

Answer

To solve the problem step by step, we will follow the principles of kinematics under constant acceleration.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The particle starts from rest, so the initial velocity \( u = 0 \, \text{m/s} \).
- The distance traveled in the first 5 seconds is given as \( x \).

2. **Use the Kinematic Equation**:
- The distance traveled under constant acceleration can be calculated using the formula:
\[ s = ut + \frac{1}{2} a t^2 \]
- For the first 5 seconds:
\[ x = 0 \cdot 5 + \frac{1}{2} a (5^2) \]
- Simplifying this gives:
\[ x = \frac{1}{2} a \cdot 25 = \frac{25a}{2} \]

3. **Calculate Total Distance in 10 Seconds**:
- Now, we need to find the total distance traveled in 10 seconds, denoted as \( y \):
\[ y = ut + \frac{1}{2} a t^2 \]
- For 10 seconds:
\[ y = 0 \cdot 10 + \frac{1}{2} a (10^2) \]
- Simplifying this gives:
\[ y = \frac{1}{2} a \cdot 100 = 50a \]

4. **Calculate Distance Traveled in the Next 5 Seconds**:
- The distance traveled in the next 5 seconds, denoted as \( d \), can be found by subtracting the distance traveled in the first 5 seconds from the total distance in 10 seconds:
\[ d = y - x \]
- Substituting the values we found:
\[ d = 50a - \frac{25a}{2} \]
- To combine these, convert \( 50a \) into a fraction:
\[ d = \frac{100a}{2} - \frac{25a}{2} = \frac{75a}{2} \]

5. **Express \( d \) in Terms of \( x \)**:
- Since we know \( x = \frac{25a}{2} \), we can express \( d \) in terms of \( x \):
\[ d = 3 \cdot \frac{25a}{2} = 3x \]

### Final Answer:
The distance traveled by the particle in the next five seconds is \( d = 3x \).

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