A man drops a ball downside from the roof of a tower of height `400` metres. At the same time another ball is thrown upside with a velocity `50 "meter"//sec` from the surface of the tower, then they will meet at which height from the surface of the tower.

Options

A. 100 meters
B. 320 meters
C. 80 meters
D. 240 meters

Explanation

To solve the problem step by step, we will analyze the motion of both balls and determine where they meet.

### Step 1: Understand the scenario
We have two balls:
- Ball A is dropped from the top of a tower of height 400 meters (initial velocity \( u_A = 0 \)).
- Ball B is thrown upwards from the ground with an initial velocity of \( u_B = 50 \, \text{m/s} \).

### Step 2: Define the equations of motion
For both balls, we can use the equations of motion under uniform acceleration due to gravity (\( g = 10 \, \text{m/s}^2 \)).

1. **For Ball A (dropped from the top)**:
The distance fallen by Ball A after time \( t \) is given by:
\[ s_A = \frac{1}{2} g t^2 = \frac{1}{2} \times 10 \times t^2 = 5t^2 \]

2. **For Ball B (thrown upwards)**:
The distance traveled by Ball B after time \( t \) is given by:
\[ s_B = u_B t - \frac{1}{2} g t^2 = 50t - 5t^2 \]

### Step 3: Set up the equation for when they meet
When the two balls meet, the sum of the distances they have traveled will equal the height of the tower:
\[ s_A + s_B = 400 \]
Substituting the expressions for \( s_A \) and \( s_B \):
\[ 5t^2 + (50t - 5t^2) = 400 \]
This simplifies to:
\[ 50t = 400 \]

### Step 4: Solve for time \( t \)
From the equation \( 50t = 400 \):
\[ t = \frac{400}{50} = 8 \, \text{seconds} \]

### Step 5: Calculate the distance traveled by Ball A
Now, we can find the distance traveled by Ball A in 8 seconds:
\[ s_A = 5t^2 = 5 \times (8^2) = 5 \times 64 = 320 \, \text{meters} \]

### Step 6: Calculate the height at which they meet
The height from the surface of the tower where they meet is:
\[ \text{Height from the surface} = 400 - s_A = 400 - 320 = 80 \, \text{meters} \]

### Final Answer
The balls will meet at a height of **80 meters** from the surface of the tower.

---