Four marbles are dropped from the top of a tower one after the other with an interval of one second. The first one reaches the ground after 4 seconds . When the first one reaches the ground the distance between the first and second, the second and third and the third and forth will be respectively

Options

A. 35, 25 and 15 m
B. 30, 20 and 10 m
C. 20, 10 and 5 m
D. 40, 30 and 20 m

Answer

To solve the problem of the distances between the marbles dropped from the tower, we will follow these steps:

### Step 1: Understand the motion of the marbles
Four marbles are dropped from a tower at one-second intervals. The first marble takes 4 seconds to reach the ground. The second marble will fall for 3 seconds, the third for 2 seconds, and the fourth for 1 second when the first marble hits the ground.

### Step 2: Use the equation of motion
The distance fallen by an object under gravity can be calculated using the formula:
\[ S = ut + \frac{1}{2} g t^2 \]
where:
- \( S \) is the distance fallen,
- \( u \) is the initial velocity (0 m/s since they are dropped),
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( t \) is the time in seconds.

### Step 3: Calculate the distance for each marble
1. **For the first marble (4 seconds):**
\[ S_1 = 0 \cdot 4 + \frac{1}{2} \cdot 10 \cdot (4^2) = 0 + \frac{1}{2} \cdot 10 \cdot 16 = 80 \, \text{meters} \]

2. **For the second marble (3 seconds):**
\[ S_2 = 0 \cdot 3 + \frac{1}{2} \cdot 10 \cdot (3^2) = 0 + \frac{1}{2} \cdot 10 \cdot 9 = 45 \, \text{meters} \]

3. **For the third marble (2 seconds):**
\[ S_3 = 0 \cdot 2 + \frac{1}{2} \cdot 10 \cdot (2^2) = 0 + \frac{1}{2} \cdot 10 \cdot 4 = 20 \, \text{meters} \]

4. **For the fourth marble (1 second):**
\[ S_4 = 0 \cdot 1 + \frac{1}{2} \cdot 10 \cdot (1^2) = 0 + \frac{1}{2} \cdot 10 \cdot 1 = 5 \, \text{meters} \]

### Step 4: Calculate the distances between the marbles
1. **Distance between the first and second marbles:**
\[ \Delta S_1 = S_1 - S_2 = 80 - 45 = 35 \, \text{meters} \]

2. **Distance between the second and third marbles:**
\[ \Delta S_2 = S_2 - S_3 = 45 - 20 = 25 \, \text{meters} \]

3. **Distance between the third and fourth marbles:**
\[ \Delta S_3 = S_3 - S_4 = 20 - 5 = 15 \, \text{meters} \]

### Final Answer
The distances between the marbles when the first one reaches the ground are:
- Between the first and second marbles: **35 meters**
- Between the second and third marbles: **25 meters**
- Between the third and fourth marbles: **15 meters**

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