Water drops fall from the roof a building 20 m high at regular time intervals. If the first drop strikes the floor when the sixth drop begins to fall, the heights of the second and fourth drops from the ground at that instant are `(g = 10 ms^(-2))`

Options

A. `12.8 m and 3.2 m`
B. `12.8 m and 7.2 m`
C. `19.2 m and 0.8 m`
D. `7.2 m and 16.8 m`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have water drops falling from a height of 20 meters at regular intervals. The first drop hits the ground when the sixth drop begins to fall. We need to find the heights of the second and fourth drops from the ground at that moment.

### Step 2: Calculate the Time Taken for the First Drop
Using the equation of motion for the first drop:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s = 20 \) m (distance fallen),
- \( u = 0 \) m/s (initial velocity),
- \( a = g = 10 \) m/s² (acceleration due to gravity),
- \( t = n \) seconds (time taken).

Substituting the values:
\[ 20 = 0 + \frac{1}{2} \times 10 \times n^2 \]
This simplifies to:
\[ 20 = 5n^2 \]
Thus,
\[ n^2 = \frac{20}{5} = 4 \quad \Rightarrow \quad n = 2 \text{ seconds} \]

### Step 3: Determine the Time Interval Between Drops
Since the first drop takes 2 seconds to hit the ground and the sixth drop begins to fall at that moment, we can find the time interval between consecutive drops. The time interval between the drops is:
\[ \text{Total time} = 2 \text{ seconds} \]
There are 5 intervals between the 6 drops, so:
\[ \text{Interval} = \frac{2}{5} \text{ seconds} = 0.4 \text{ seconds} \]

### Step 4: Calculate the Time for the Second Drop
The second drop falls for:
\[ t_2 = n - \text{Interval} = 2 - 0.4 = 1.6 \text{ seconds} \]

### Step 5: Calculate the Height of the Second Drop
Using the equation of motion again for the second drop:
\[ s_2 = ut_2 + \frac{1}{2} g t_2^2 \]
Substituting the values:
\[ s_2 = 0 + \frac{1}{2} \times 10 \times (1.6)^2 \]
Calculating:
\[ s_2 = 5 \times 2.56 = 12.8 \text{ meters} \]
The height of the second drop from the ground is:
\[ \text{Height from ground} = 20 - 12.8 = 7.2 \text{ meters} \]

### Step 6: Calculate the Time for the Fourth Drop
The fourth drop falls for:
\[ t_4 = n - 3 \times \text{Interval} = 2 - 3 \times 0.4 = 2 - 1.2 = 0.8 \text{ seconds} \]

### Step 7: Calculate the Height of the Fourth Drop
Using the equation of motion for the fourth drop:
\[ s_4 = ut_4 + \frac{1}{2} g t_4^2 \]
Substituting the values:
\[ s_4 = 0 + \frac{1}{2} \times 10 \times (0.8)^2 \]
Calculating:
\[ s_4 = 5 \times 0.64 = 3.2 \text{ meters} \]
The height of the fourth drop from the ground is:
\[ \text{Height from ground} = 20 - 3.2 = 16.8 \text{ meters} \]

### Final Answers
- Height of the second drop from the ground: **7.2 meters**
- Height of the fourth drop from the ground: **16.8 meters**