Two bodies of masses 4 kg and 16 kg at rest to acted upon by same force. The ratio of times required to attain the same speed is,

Options

A. `1 : 1`
B. `4 : 1`
C. `1 : 4`
D. `1 :2`

Answer

To solve the problem, we need to find the ratio of times required for two bodies of different masses to attain the same speed when acted upon by the same force. Let's break down the solution step by step.

### Step 1: Identify the masses and the force
Let:
- Mass of the first body, \( m_1 = 4 \, \text{kg} \)
- Mass of the second body, \( m_2 = 16 \, \text{kg} \)
- Force applied, \( F \) (same for both bodies)

### Step 2: Calculate the acceleration for each body
Using Newton's second law, the acceleration \( a \) can be calculated as:
\[ a = \frac{F}{m} \]
For the first body:
\[ a_1 = \frac{F}{m_1} = \frac{F}{4} \]
For the second body:
\[ a_2 = \frac{F}{m_2} = \frac{F}{16} \]

### Step 3: Use the kinematic equation to find time
Since both bodies start from rest, we can use the equation:
\[ v = u + at \]
where \( u = 0 \) (initial velocity), \( v \) is the final velocity, and \( t \) is the time taken.

For the first body:
\[ v = 0 + a_1 t_1 \implies v = \frac{F}{4} t_1 \implies t_1 = \frac{4v}{F} \]

For the second body:
\[ v = 0 + a_2 t_2 \implies v = \frac{F}{16} t_2 \implies t_2 = \frac{16v}{F} \]

### Step 4: Find the ratio of times
Now, we need to find the ratio \( \frac{t_1}{t_2} \):
\[ \frac{t_1}{t_2} = \frac{\frac{4v}{F}}{\frac{16v}{F}} = \frac{4v}{F} \cdot \frac{F}{16v} = \frac{4}{16} = \frac{1}{4} \]

### Conclusion
The ratio of times required to attain the same speed is:
\[ \frac{t_1}{t_2} = 1:4 \]