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, we will use the equations of motion under constant acceleration. Here’s a step-by-step solution:

### Step 1: Understand the initial conditions
The particle starts from rest, which means the initial velocity \( u = 0 \). The distance traveled in the first 5 seconds is given as \( x \).

### Step 2: Use the equation of motion
The equation of motion for distance traveled under constant acceleration is:
\[ S = ut + \frac{1}{2} a t^2 \]
For the first 5 seconds:
- \( S = x \)
- \( u = 0 \)
- \( t = 5 \) seconds

Substituting these values into the equation gives:
\[ x = 0 \cdot 5 + \frac{1}{2} a (5^2) \]
This simplifies to:
\[ x = \frac{1}{2} a \cdot 25 \]
\[ x = \frac{25a}{2} \]

### Step 3: Solve for acceleration \( a \)
From the equation \( x = \frac{25a}{2} \), we can solve for \( a \):
\[ a = \frac{2x}{25} \]

### Step 4: Calculate distance traveled in the next 5 seconds
Now, we need to find the distance traveled during the next 5 seconds (from \( t = 5 \) seconds to \( t = 10 \) seconds). The total distance traveled in 10 seconds can be calculated using the same equation:
\[ S_{10} = ut + \frac{1}{2} a t^2 \]
For \( t = 10 \) seconds:
\[ S_{10} = 0 \cdot 10 + \frac{1}{2} a (10^2) \]
Substituting \( a = \frac{2x}{25} \):
\[ S_{10} = \frac{1}{2} \cdot \frac{2x}{25} \cdot 100 \]
This simplifies to:
\[ S_{10} = \frac{100x}{25} = 4x \]

### Step 5: Calculate distance traveled in the next 5 seconds
The distance traveled in the next 5 seconds (from \( t = 5 \) seconds to \( t = 10 \) seconds) is:
\[ S_{next} = S_{10} - S_{5} \]
Where \( S_{5} = x \) (distance traveled in the first 5 seconds):
\[ S_{next} = 4x - x = 3x \]

### Final Answer
The distance traveled by the particle in the next 5 seconds is \( 3x \).

---