(A) : On the rough horizontal surface if the external force is doubled then the acceleration also becomes more than doubled
(R) : For a small velocities sliding friction between a body and surface is constant
(a) Both (A) and (R) are true and (R) is the correct explanation of (A)
(b) Both (A) and (R) are true and (R) is not the correct explanation of (A)
(c) (A) is true but (R) is false
(d) Both (A) and (R) are false

Options

A. Both (A) and (R) are true and (R) is the correct explanation of (A)
B. Both (A) and (R) are true and (R) is not the
C. (A) is true but ( R) is false
D. Both (A) and (R ) are false

Answer

To solve the assertion and reason question, we need to analyze both statements separately.

### Step 1: Analyze the Assertion (A)
The assertion states: "On the rough horizontal surface, if the external force is doubled, then the acceleration also becomes more than doubled."

1. **Understanding the Forces**:
- Let the mass of the object be \( m \).
- The coefficient of friction between the object and the surface is \( \mu \).
- The normal force \( n \) is equal to \( mg \) (where \( g \) is the acceleration due to gravity).

2. **Applying Newton's Second Law**:
- The frictional force \( f \) opposing the motion is given by \( f = \mu n = \mu mg \).
- When an external force \( F \) is applied, the net force acting on the object is \( F - f \).
- Therefore, the acceleration \( A_1 \) can be expressed as:
\[ A_1 = \frac{F - \mu mg}{m} \]

3. **Doubling the External Force**:
- If the external force is doubled, the new force becomes \( 2F \).
- The new acceleration \( A_2 \) is given by:
\[ A_2 = \frac{2F - \mu mg}{m} \]

4. **Comparing Accelerations**:
- To check if \( A_2 \) is more than double \( A_1 \):
\[ A_2 = \frac{2F - \mu mg}{m} \quad \text{and} \quad 2A_1 = 2 \left( \frac{F - \mu mg}{m} \right) = \frac{2F - 2\mu mg}{m} \]
- Clearly, \( A_2 \) is not more than \( 2A_1 \) because the term \( -\mu mg \) does not change in proportion when doubling the force. Thus, the assertion is **false**.

### Step 2: Analyze the Reason (R)
The reason states: "For small velocities, sliding friction between a body and surface is constant."

1. **Understanding Sliding Friction**:
- Sliding friction (kinetic friction) is represented by \( f_k = \mu_k n \).
- The coefficient of kinetic friction \( \mu_k \) is generally considered constant for a given surface and does not depend on the velocity of the object.

2. **Conclusion about the Reason**:
- Since kinetic friction remains constant regardless of the velocity (whether small or large), the statement is also **false**.

### Final Conclusion
Both the assertion (A) and the reason (R) are false. Therefore, the correct answer is:
**(d) Both (A) and (R) are false.**