A particle starting from rest falls from a certain height. Assuming that the acceleration due to gravity remain the same throughout the motion, its displacements in three successive half second intervals are `S_(1), S_(2) and S_(3)` then

Options

A. `S_(1): S_(2): S_(3) = 1 : 5 : 9`
B. `S_(1) : S_(2) : S_(3) = 1: 3:5`
C. `S_(1) : S_(2): S_(3) = 9:2:3`
D. `S_(1): S_(2) : S_(3) = 1: 1: 1`

Explanation

To solve the problem of a particle falling from rest under the influence of gravity, we need to find the displacements \( S_1 \), \( S_2 \), and \( S_3 \) during three successive half-second intervals. Let's break this down step by step.

### Step 1: Understand the Motion
The particle starts from rest, meaning its initial velocity \( u = 0 \). The acceleration due to gravity \( g \) is constant throughout the motion. We will use the second equation of motion:
\[ S = ut + \frac{1}{2} g t^2 \]

### Step 2: Calculate \( S_1 \)
For the first half-second interval (from \( t = 0 \) to \( t = 0.5 \) seconds):
- Initial velocity \( u = 0 \)
- Time \( t = 0.5 \) seconds

Using the equation:
\[ S_1 = 0 \cdot (0.5) + \frac{1}{2} g (0.5)^2 = \frac{1}{2} g \cdot \frac{1}{4} = \frac{g}{8} \]

### Step 3: Calculate \( S_2 \)
For the second half-second interval (from \( t = 0.5 \) to \( t = 1.0 \) seconds):
- The total time is now \( t = 1.0 \) seconds.
- The displacement from the start to \( t = 1.0 \) seconds is:
\[ S_{total} = \frac{1}{2} g (1.0)^2 = \frac{g}{2} \]
- The displacement \( S_2 \) is the total displacement at \( t = 1.0 \) seconds minus the displacement at \( t = 0.5 \) seconds:
\[ S_2 = S_{total} - S_1 = \frac{g}{2} - \frac{g}{8} = \frac{4g}{8} - \frac{g}{8} = \frac{3g}{8} \]

### Step 4: Calculate \( S_3 \)
For the third half-second interval (from \( t = 1.0 \) to \( t = 1.5 \) seconds):
- The total time is \( t = 1.5 \) seconds.
- The displacement from the start to \( t = 1.5 \) seconds is:
\[ S_{total} = \frac{1}{2} g (1.5)^2 = \frac{1}{2} g \cdot \frac{9}{4} = \frac{9g}{8} \]
- The displacement \( S_3 \) is the total displacement at \( t = 1.5 \) seconds minus the displacement at \( t = 1.0 \) seconds:
\[ S_3 = S_{total} - (S_1 + S_2) = \frac{9g}{8} - \left(\frac{g}{8} + \frac{3g}{8}\right) = \frac{9g}{8} - \frac{4g}{8} = \frac{5g}{8} \]

### Step 5: Find the Relationship Between \( S_1, S_2, S_3 \)
Now we have:
- \( S_1 = \frac{g}{8} \)
- \( S_2 = \frac{3g}{8} \)
- \( S_3 = \frac{5g}{8} \)

To find the ratio \( S_1 : S_2 : S_3 \):
\[ S_1 : S_2 : S_3 = \frac{g}{8} : \frac{3g}{8} : \frac{5g}{8} = 1 : 3 : 5 \]

### Final Answer
The relationship between the displacements is:
\[ S_1 : S_2 : S_3 = 1 : 3 : 5 \]