Equal net forces act on two different block (A) and (B) masses (m) and 4(m) respectively For same displacement, identify the correct statement.

Options

A. Their kinetic energies are in the ratio `(K_(A))/(K_(B)) = (1)/(4)`
B. Their speeds are in the ration `(v_(A))/(v_(B)) = (1)/(1)`
C. Work done on the block are in the ratio `(W_(A))/(W_(B)) = (1)/(1)`
D. All of the above

Explanation

To solve the problem, we need to analyze the situation where equal net forces act on two blocks (A and B) with masses \( m \) and \( 4m \) respectively, for the same displacement. We will derive the relationships between work done, kinetic energy, and speed for both blocks.

### Step 1: Understand the Work Done
The work done (W) on an object is given by the formula:
\[ W = F \cdot S \]
where \( F \) is the force applied and \( S \) is the displacement. Since equal net forces act on both blocks and the displacement is the same, we can denote the force as \( F \) and the displacement as \( S \).

### Step 2: Calculate Work Done on Both Blocks
For both blocks A and B:
- Work done on block A, \( W_A = F \cdot S \)
- Work done on block B, \( W_B = F \cdot S \)

Since both forces and displacements are equal:
\[ W_A = W_B \]
Thus, the ratio of work done on block A to block B is:
\[ \frac{W_A}{W_B} = 1 : 1 \]

### Step 3: Apply the Work-Energy Theorem
According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. Therefore, we can write:
\[ W_A = KE_A \quad \text{and} \quad W_B = KE_B \]
Since \( W_A = W_B \), we have:
\[ KE_A = KE_B \]
This implies:
\[ \frac{KE_A}{KE_B} = 1 : 1 \]

### Step 4: Relate Kinetic Energy to Speed
The kinetic energy (KE) of an object is given by:
\[ KE = \frac{1}{2} m v^2 \]
For block A:
\[ KE_A = \frac{1}{2} m v_A^2 \]
For block B:
\[ KE_B = \frac{1}{2} (4m) v_B^2 = 2m v_B^2 \]
Setting the kinetic energies equal gives:
\[ \frac{1}{2} m v_A^2 = 2m v_B^2 \]
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ \frac{1}{2} v_A^2 = 2 v_B^2 \]
Rearranging gives:
\[ v_A^2 = 4 v_B^2 \]
Taking the square root:
\[ \frac{v_A}{v_B} = 2 : 1 \]

### Conclusion
From our analysis:
- The work done on both blocks is in the ratio \( 1 : 1 \).
- The kinetic energy of both blocks is also in the ratio \( 1 : 1 \).
- The speeds of the blocks are in the ratio \( 2 : 1 \).

Thus, the correct statement is that the work done on the blocks is in the ratio \( 1 : 1 \).

### Final Answer
The correct statement is that the work done on blocks A and B is in the ratio \( 1 : 1 \).