Same force acts on two bodies of different masses 3 kg and 5 kg initially at rest. The ratio of time required to acquire same final velocity is

Options

A. `5:3`
B. `25:9`
C. ` 9:25`
D. `3:5`

Explanation

To solve the problem of determining the ratio of time required for two bodies of different masses (3 kg and 5 kg) to acquire the same final velocity under the influence of the same force, we can follow these steps:

### Step 1: Understand the relationship between force, mass, acceleration, and time.
According to Newton's second law of motion, the force acting on an object is equal to the mass of the object multiplied by its acceleration:
\[ F = m \cdot a \]
Where:
- \( F \) is the force,
- \( m \) is the mass,
- \( a \) is the acceleration.

### Step 2: Express acceleration in terms of final velocity and time.
Acceleration can also be defined as the change in velocity over time:
\[ a = \frac{v - u}{t} \]
Since both bodies start from rest, the initial velocity \( u = 0 \). Therefore, the equation simplifies to:
\[ a = \frac{v}{t} \]

### Step 3: Set up the equations for both masses.
For the first body (mass = 3 kg):
\[ F = 3 \cdot a_1 \]
Substituting for acceleration:
\[ F = 3 \cdot \frac{v}{t_1} \]
Thus:
\[ F = \frac{3v}{t_1} \]
(1)

For the second body (mass = 5 kg):
\[ F = 5 \cdot a_2 \]
Substituting for acceleration:
\[ F = 5 \cdot \frac{v}{t_2} \]
Thus:
\[ F = \frac{5v}{t_2} \]
(2)

### Step 4: Set the equations equal to each other.
Since the force \( F \) is the same for both bodies, we can set equations (1) and (2) equal to each other:
\[ \frac{3v}{t_1} = \frac{5v}{t_2} \]

### Step 5: Simplify the equation.
We can cancel \( v \) from both sides (assuming \( v \neq 0 \)):
\[ \frac{3}{t_1} = \frac{5}{t_2} \]

### Step 6: Cross-multiply to find the ratio of times.
Cross-multiplying gives:
\[ 3t_2 = 5t_1 \]

### Step 7: Rearrange to find the ratio \( \frac{t_1}{t_2} \).
Rearranging the equation:
\[ \frac{t_1}{t_2} = \frac{3}{5} \]

### Conclusion
The ratio of the time required for the two bodies to acquire the same final velocity is:
\[ t_1 : t_2 = 3 : 5 \]