Three particles are projected upwards with initial speeds `10 m//s,20 m//s,30 m//s`. The displacements covered by them in their last second of motion are `x_(1),x_(2),x_(3)` then:

Options

A. `x_(1):x_(2):x_(3)=1:2:3`
B. `x_(1):x_(2):x_(3)=1:4:9`
C. `x_(1):x_(2):x_(3)=1:5:7`
D. none of these

Explanation

To solve the problem, we need to find the displacements \( x_1, x_2, x_3 \) covered by three particles projected upwards with initial speeds \( u_1 = 10 \, \text{m/s} \), \( u_2 = 20 \, \text{m/s} \), and \( u_3 = 30 \, \text{m/s} \) during their last second of motion.

### Step 1: Understand the motion
When a particle is projected upwards, it will eventually stop momentarily at its highest point before falling back down. The displacement covered in the last second of motion can be calculated using the equations of motion.

### Step 2: Use the equation of motion
The displacement \( s \) during the last second of motion can be calculated using the formula:
\[ s = u \cdot t - \frac{1}{2} g t^2 \]
where:
- \( u \) is the initial velocity,
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( t \) is the time interval (which is \( 1 \, \text{s} \) for the last second).

### Step 3: Calculate \( x_1 \) for the first particle
For the first particle with \( u_1 = 10 \, \text{m/s} \):
\[ x_1 = u_1 \cdot 1 - \frac{1}{2} g \cdot (1)^2 \]
\[ x_1 = 10 \cdot 1 - \frac{1}{2} \cdot 10 \cdot 1^2 \]
\[ x_1 = 10 - 5 = 5 \, \text{m} \]

### Step 4: Calculate \( x_2 \) for the second particle
For the second particle with \( u_2 = 20 \, \text{m/s} \):
\[ x_2 = u_2 \cdot 1 - \frac{1}{2} g \cdot (1)^2 \]
\[ x_2 = 20 \cdot 1 - \frac{1}{2} \cdot 10 \cdot 1^2 \]
\[ x_2 = 20 - 5 = 15 \, \text{m} \]

### Step 5: Calculate \( x_3 \) for the third particle
For the third particle with \( u_3 = 30 \, \text{m/s} \):
\[ x_3 = u_3 \cdot 1 - \frac{1}{2} g \cdot (1)^2 \]
\[ x_3 = 30 \cdot 1 - \frac{1}{2} \cdot 10 \cdot 1^2 \]
\[ x_3 = 30 - 5 = 25 \, \text{m} \]

### Step 6: Find the ratio \( x_1 : x_2 : x_3 \)
Now we have:
- \( x_1 = 5 \, \text{m} \)
- \( x_2 = 15 \, \text{m} \)
- \( x_3 = 25 \, \text{m} \)

The ratio \( x_1 : x_2 : x_3 \) is:
\[ x_1 : x_2 : x_3 = 5 : 15 : 25 \]
Dividing each term by 5 gives:
\[ 1 : 3 : 5 \]

### Final Answer
The ratio of displacements covered by the three particles in their last second of motion is:
\[ \boxed{1 : 3 : 5} \]