Two balls of equal masses are thrown upwards, along the same vertical direction at an interval of 2 seconds, with the same initial velocity of `40m//s`. Then these collide at a height of (Take `g=10m//s^(2)`).

Options

A. `120m`
B. `75m`
C. `200m`
D. `45m`

Answer

To solve the problem of two balls thrown upwards and colliding at a certain height, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two balls of equal mass, both thrown upwards with an initial velocity of \( u = 40 \, \text{m/s} \).
- The first ball is thrown at \( t = 0 \) seconds, and the second ball is thrown at \( t = 2 \) seconds.
- We need to find the height at which they collide.

2. **Using the Equation of Motion**:
- The height \( h \) of an object thrown upwards can be calculated using the formula:
\[ h = ut - \frac{1}{2}gt^2 \]
- Here, \( g \) (acceleration due to gravity) is given as \( 10 \, \text{m/s}^2 \).

3. **Finding the Height of the First Ball**:
- For the first ball, which is thrown at \( t = 0 \):
- At time \( t_1 \), the height is:
\[ h_1 = 40t_1 - \frac{1}{2} \cdot 10 \cdot t_1^2 \]
- This simplifies to:
\[ h_1 = 40t_1 - 5t_1^2 \]

4. **Finding the Height of the Second Ball**:
- The second ball is thrown at \( t = 2 \) seconds, so its height at time \( t_2 \) (where \( t_2 = t_1 - 2 \)) is:
- At time \( t_1 \):
\[ h_2 = 40(t_1 - 2) - \frac{1}{2} \cdot 10 \cdot (t_1 - 2)^2 \]
- This simplifies to:
\[ h_2 = 40(t_1 - 2) - 5(t_1^2 - 4t_1 + 4) \]
- Expanding this gives:
\[ h_2 = 40t_1 - 80 - 5t_1^2 + 20t_1 - 20 \]
- Thus:
\[ h_2 = -5t_1^2 + 60t_1 - 100 \]

5. **Setting the Heights Equal**:
- Since both balls collide at the same height, we set \( h_1 = h_2 \):
\[ 40t_1 - 5t_1^2 = -5t_1^2 + 60t_1 - 100 \]
- Simplifying this gives:
\[ 40t_1 = 60t_1 - 100 \]
- Rearranging:
\[ 20t_1 = 100 \implies t_1 = 5 \, \text{seconds} \]

6. **Calculating the Height at Collision**:
- Now substituting \( t_1 = 5 \) seconds back into the equation for height:
\[ h = 40(5) - \frac{1}{2} \cdot 10 \cdot (5^2) \]
- This simplifies to:
\[ h = 200 - 125 = 75 \, \text{meters} \]

### Final Answer:
The two balls collide at a height of **75 meters**.