(A) : When external force acting on a body moving on a rough horizontal surface is doubled, the acceleration of the body is doubled.
(R ) : The acceleration of a body moving on a rough horizontal surface is proportional to the resultant force acting on it.

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 correct explanation of (A)
C. (A) is true but (R ) is false
D. (A) is false but (R ) is true

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question.

### Step 1: Understand the Assertion
The assertion states that when the external force acting on a body moving on a rough horizontal surface is doubled, the acceleration of the body is also doubled.

### Step 2: Analyze the Forces
When a force \( F \) is applied to a body on a rough horizontal surface, the body experiences a frictional force \( f \) opposing its motion. The net force \( F_{net} \) acting on the body can be expressed as:
\[ F_{net} = F - f \]
where \( f \) is the frictional force.

### Step 3: Write the Equation of Motion
According to Newton's second law, the acceleration \( a \) of the body can be expressed as:
\[ a = \frac{F_{net}}{m} = \frac{F - f}{m} \]
where \( m \) is the mass of the body.

### Step 4: Doubling the External Force
Now, if the external force is doubled, the new force becomes \( 2F \). The new acceleration \( a' \) can be calculated as:
\[ a' = \frac{2F - f}{m} \]

### Step 5: Compare the Accelerations
To see if the acceleration doubles, we need to compare \( a' \) with \( 2a \):
\[ 2a = 2 \left( \frac{F - f}{m} \right) = \frac{2F - 2f}{m} \]

### Step 6: Analyze the Frictional Force
The frictional force \( f \) is typically given by:
\[ f = \mu N \]
where \( \mu \) is the coefficient of friction and \( N \) is the normal force. On a horizontal surface, \( N = mg \), so:
\[ f = \mu mg \]
This frictional force remains constant as long as the applied force does not exceed the maximum static friction.

### Step 7: Conclusion on Assertion
Since the frictional force \( f \) does not change when the applied force is doubled (as long as it does not exceed the maximum static friction), we find that:
\[ a' = \frac{2F - f}{m} \neq 2 \left( \frac{F - f}{m}\right) \]
Thus, the assertion is **false**.

### Step 8: Understand the Reason
The reason states that the acceleration of a body moving on a rough horizontal surface is proportional to the resultant force acting on it. This is indeed true because:
\[ a = \frac{F - f}{m} \]
shows that acceleration is directly related to the net force acting on the body.

### Final Conclusion
- **Assertion (A)**: False
- **Reason (R)**: True

### Correct Option
The correct option is that the assertion is false, but the reason is true.

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