Thre stones A, B and C are simultaneously projected from same point with same speed. A is thrown upwards, B is thrown horizontally and C is thrown downwards from a boulding. When the distance between stone A and C becomes 10 m, then distance between A and B will be-

Options

A. 10 m
B. 5 m
C. `5sqrt2m`
D. `10sqrt2m`

Explanation

To solve the problem, we need to analyze the motion of the three stones A, B, and C, which are projected from the same point with the same speed. Here's a step-by-step solution:

### Step 1: Understand the motion of each stone
- **Stone A** is thrown upwards with an initial velocity \( u \).
- **Stone B** is thrown horizontally with the same initial velocity \( u \).
- **Stone C** is thrown downwards with the same initial velocity \( u \).

### Step 2: Analyze the vertical motion of stones A and C
- The vertical motion of stone A (upwards) is influenced by gravity, which acts downwards with an acceleration \( g \).
- The vertical motion of stone C (downwards) is also influenced by gravity, acting downwards with acceleration \( g \).
- At any time \( t \), the height of stone A above the ground can be given by:
\[ h_A = ut - \frac{1}{2}gt^2 \]
- The height of stone C below the starting point can be given by:
\[ h_C = ut + \frac{1}{2}gt^2 \]

### Step 3: Determine the distance between stones A and C
- The total distance between stones A and C when they are 10 m apart can be expressed as:
\[ d_{AC} = h_A + h_C = \left(ut - \frac{1}{2}gt^2\right) + \left(ut + \frac{1}{2}gt^2\right) = 2ut \]
- We know that \( d_{AC} = 10 \, \text{m} \), so:
\[ 2ut = 10 \implies ut = 5 \, \text{m} \]

### Step 4: Analyze the horizontal motion of stone B
- Stone B moves horizontally with a constant speed \( u \).
- The horizontal distance covered by stone B after time \( t \) is:
\[ d_B = ut \]
- Since we found \( ut = 5 \, \text{m} \), we have:
\[ d_B = 5 \, \text{m} \]

### Step 5: Calculate the distance between stones A and B
- At time \( t \), stone A is at a height \( h_A \) and stone B is at a horizontal distance \( d_B \).
- The distance \( d_{AB} \) between stones A and B can be calculated using the Pythagorean theorem:
\[ d_{AB} = \sqrt{(d_B)^2 + (h_A)^2} \]
- We know \( d_B = 5 \, \text{m} \) and \( h_A = 5 \, \text{m} \) (as derived from the vertical motion of stone A when the distance between A and C is 10 m):
\[ d_{AB} = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \, \text{m} \]

### Conclusion
Thus, the distance between stone A and stone B when the distance between stone A and stone C is 10 m is:
\[ \boxed{5\sqrt{2} \, \text{m}} \]