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. 50 m
B. 75 m
C. 100 m
D. 125 m

Explanation

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

### Step 1: Understand the problem
Two balls of equal mass are thrown upwards with the same initial velocity of 40 m/s, with a time interval of 2 seconds between their throws. We need to find the height at which they collide.

### Step 2: Define the equations of motion
The displacement \( s \) of an object under uniform acceleration can be given by the equation:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( u \) = initial velocity
- \( a \) = acceleration (in this case, \( -g \) because gravity acts downwards)
- \( t \) = time

For our problem, the acceleration \( a = -10 \, \text{m/s}^2 \) (since \( g = 10 \, \text{m/s}^2 \)).

### Step 3: Set up the equations for both balls
Let:
- \( t \) be the time taken by the first ball to reach the collision point.
- The second ball is thrown 2 seconds later, so it will take \( t - 2 \) seconds to reach the same point.

For the first ball (Ball 1):
\[ s_1 = 40t - \frac{1}{2} \cdot 10 \cdot t^2 = 40t - 5t^2 \]

For the second ball (Ball 2):
\[ s_2 = 40(t - 2) - \frac{1}{2} \cdot 10 \cdot (t - 2)^2 \]
Expanding \( s_2 \):
\[ s_2 = 40(t - 2) - 5(t - 2)^2 \]
\[ s_2 = 40t - 80 - 5(t^2 - 4t + 4) \]
\[ s_2 = 40t - 80 - 5t^2 + 20t - 20 \]
\[ s_2 = 60t - 5t^2 - 100 \]

### Step 4: Set the displacements equal to each other
Since both balls collide at the same height:
\[ s_1 = s_2 \]
Thus:
\[ 40t - 5t^2 = 60t - 5t^2 - 100 \]

### Step 5: Simplify the equation
Cancel \( -5t^2 \) from both sides:
\[ 40t = 60t - 100 \]
Rearranging gives:
\[ 100 = 60t - 40t \]
\[ 100 = 20t \]
\[ t = 5 \, \text{seconds} \]

### Step 6: Calculate the height at which they collide
Now, substitute \( t = 5 \) seconds back into the equation for \( s_1 \):
\[ s_1 = 40(5) - 5(5^2) \]
\[ s_1 = 200 - 5 \cdot 25 \]
\[ s_1 = 200 - 125 \]
\[ s_1 = 75 \, \text{meters} \]

### Conclusion
The balls collide at a height of **75 meters**.