Water drips from the nozzle of a shower onto the floor 200 cm below. The drops fall at regular ( equal) intervals of time, the first drop striking the floor at the instant the fourth drop begins to fall. When the first drop strikes the floor, how far below the nozzle are the (a) second and (b) third drops ?

Answer

To solve the problem step by step, we need to analyze the motion of the water drops falling from the shower nozzle. Here’s how we can approach the question:

### Given Data:
- Height of the nozzle from the floor, \( h = 200 \, \text{cm} \)
- The drops fall at equal intervals of time.
- The first drop strikes the floor when the fourth drop begins to fall.

### Step 1: Understand the Time Intervals
Let the time interval between the drops be \( T \).
- The first drop takes \( 3T \) to reach the floor.
- The second drop takes \( 2T \).
- The third drop takes \( T \).
- The fourth drop is just about to fall.

### Step 2: Use the Equation of Motion
We will use the equation of motion for free fall:
\[ S = ut + \frac{1}{2} a t^2 \]
Where:
- \( S \) is the displacement,
- \( u \) is the initial velocity (which is 0 for all drops),
- \( a \) is the acceleration due to gravity \( g \) (approximately \( 980 \, \text{cm/s}^2 \)),
- \( t \) is the time.

For the first drop:
\[ 200 = 0 \cdot (3T) + \frac{1}{2} g (3T)^2 \]
This simplifies to:
\[ 200 = \frac{1}{2} g (9T^2) \]
\[ 200 = \frac{9gT^2}{2} \]
\[ T^2 = \frac{400}{9g} \]

### Step 3: Calculate the Displacement for the Second and Third Drops
**For the second drop:**
\[ S_2 = \frac{1}{2} g (2T)^2 \]
Substituting \( T^2 \):
\[ S_2 = \frac{1}{2} g (4T^2) = 2gT^2 \]
Substituting \( T^2 \):
\[ S_2 = 2g \left(\frac{400}{9g}\right) = \frac{800}{9} \, \text{cm} \approx 88.89 \, \text{cm} \]

**For the third drop:**
\[ S_3 = \frac{1}{2} g (T)^2 \]
Substituting \( T^2 \):
\[ S_3 = \frac{1}{2} g (T^2) = \frac{1}{2} g \left(\frac{400}{9g}\right) = \frac{200}{9} \, \text{cm} \approx 22.22 \, \text{cm} \]

### Final Answers:
(a) The distance of the second drop below the nozzle is approximately \( 88.89 \, \text{cm} \).

(b) The distance of the third drop below the nozzle is approximately \( 22.22 \, \text{cm} \).