Two bodies of masses 4 kg and 16 kg ar at rest. The ratio of times for which the same force must act on them to produce the same kinetic energy in both of them is

Options

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

Answer

To solve the problem step by step, we need to find the ratio of the times for which the same force must act on two bodies of masses 4 kg and 16 kg to produce the same kinetic energy in both of them.

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

### Step 2: Set up the equations for both bodies
Let:
- \( m_1 = 4 \, \text{kg} \) (mass of the first body)
- \( m_2 = 16 \, \text{kg} \) (mass of the second body)
- \( F \) = same force acting on both bodies
- \( a_1 \) = acceleration of the first body
- \( a_2 \) = acceleration of the second body

From Newton's second law, we can write:
\[ F = m_1 \cdot a_1 \]
\[ F = m_2 \cdot a_2 \]

### Step 3: Relate the accelerations
Since the same force \( F \) is acting on both bodies:
\[ m_1 \cdot a_1 = m_2 \cdot a_2 \]
This implies:
\[ a_1 = \frac{m_2}{m_1} \cdot a_2 \]

### Step 4: Calculate the ratio of accelerations
Substituting the values of masses:
\[ a_1 = \frac{16}{4} \cdot a_2 = 4 \cdot a_2 \]
Thus, we have:
\[ a_1 = 4a_2 \]

### Step 5: Use the kinetic energy formula
The kinetic energy (KE) of an object is given by:
\[ KE = \frac{1}{2} m v^2 \]
We want both bodies to have the same kinetic energy after the same time \( t \).

### Step 6: Express velocity in terms of acceleration and time
The velocity of each body after time \( t \) can be expressed as:
\[ v_1 = a_1 \cdot t \]
\[ v_2 = a_2 \cdot t \]

### Step 7: Set the kinetic energies equal
Setting the kinetic energies equal gives:
\[ \frac{1}{2} m_1 (a_1 t)^2 = \frac{1}{2} m_2 (a_2 t)^2 \]
This simplifies to:
\[ m_1 (a_1 t)^2 = m_2 (a_2 t)^2 \]

### Step 8: Substitute for \( a_1 \)
Substituting \( a_1 = 4a_2 \) into the equation:
\[ m_1 (4a_2 t)^2 = m_2 (a_2 t)^2 \]
This simplifies to:
\[ m_1 \cdot 16 a_2^2 t^2 = m_2 \cdot a_2^2 t^2 \]
Dividing both sides by \( a_2^2 t^2 \) (assuming \( a_2 \) and \( t \) are not zero):
\[ m_1 \cdot 16 = m_2 \]

### Step 9: Substitute the masses
Substituting \( m_1 = 4 \, \text{kg} \) and \( m_2 = 16 \, \text{kg} \):
\[ 4 \cdot 16 = 16 \]
This confirms the relationship holds.

### Step 10: Find the ratio of times
Since \( a_1 = 4a_2 \), the time taken for both bodies to reach the same kinetic energy can be expressed as:
\[ t_1 = \frac{v_1}{a_1} = \frac{4a_2 t}{4a_2} = t \]
\[ t_2 = \frac{v_2}{a_2} = \frac{a_2 t}{a_2} = t \]

Thus, the ratio of times \( t_1 : t_2 \) is:
\[ t_1 : t_2 = 1 : 1 \]

### Final Answer
The ratio of times for which the same force must act on them to produce the same kinetic energy in both of them is \( 1 : 1 \).