A ball falls from height `h`. After 1 s, another ball falls freely from a point `25 m` below the point from where the first ball falls. Both of them reach the ground at the same time. The value of `h` is

Options

A. 30 m
B. 45 m
C. 60 m
D. 75 m

Explanation

To solve the problem step by step, we can break it down as follows:

### Step 1: Understand the Problem
We have two balls:
- The first ball falls from a height `h` and starts falling at time `t = 0`.
- The second ball falls from a point 25 meters below the first ball and starts falling after 1 second (at `t = 1` second).
- Both balls reach the ground at the same time.

### Step 2: Define Variables
Let:
- `t` = time taken by the first ball to reach the ground.
- The second ball will take `t - 1` seconds to reach the ground (since it starts falling 1 second later).

### Step 3: Write the Equations of Motion
For the first ball:
- The distance fallen by the first ball is `h`, and using the equation of motion:
\[ h = \frac{1}{2} g t^2 \]
where \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity).

For the second ball:
- The distance fallen by the second ball is \( h - 25 \) meters, and it falls for \( t - 1 \) seconds:
\[ h - 25 = \frac{1}{2} g (t - 1)^2 \]

### Step 4: Substitute the Value of g
Substituting \( g = 10 \):
1. From the first ball:
\[ h = \frac{1}{2} \cdot 10 \cdot t^2 = 5t^2 \]
2. From the second ball:
\[ h - 25 = \frac{1}{2} \cdot 10 \cdot (t - 1)^2 = 5(t - 1)^2 \]

### Step 5: Expand the Equation for the Second Ball
Expanding the equation for the second ball:
\[ h - 25 = 5(t^2 - 2t + 1) = 5t^2 - 10t + 5 \]
Thus, we can write:
\[ h = 5t^2 - 10t + 30 \]

### Step 6: Set the Two Expressions for h Equal
Now, we have two expressions for \( h \):
1. \( h = 5t^2 \)
2. \( h = 5t^2 - 10t + 30 \)

Setting them equal to each other:
\[ 5t^2 = 5t^2 - 10t + 30 \]

### Step 7: Simplify the Equation
Cancelling \( 5t^2 \) from both sides:
\[ 0 = -10t + 30 \]
Rearranging gives:
\[ 10t = 30 \implies t = 3 \, \text{seconds} \]

### Step 8: Calculate the Height h
Now substitute \( t = 3 \) back into the equation for \( h \):
\[ h = 5t^2 = 5(3^2) = 5 \cdot 9 = 45 \, \text{meters} \]

### Final Answer
The value of \( h \) is **45 meters**.

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