Block A is moving with acceleration A along a frictionless horizontal surface. When a second block, B is placed on top of Block A the acceleration of the combined blocks drops to 1/5 the original value. What is the ratio of the mass of A to the mass of B?

Options

A. `5:1`
B. `1:4`
C. `3:1`
D. `2:1`

Explanation

To solve the problem, we need to analyze the situation using Newton's second law of motion. Let's break it down step by step.

### Step 1: Understand the initial conditions
Block A is moving with an acceleration \( A \) on a frictionless surface. The force acting on Block A can be expressed as:
\[ F = m_A \cdot A \]
where \( m_A \) is the mass of Block A.

### Step 2: Introduce Block B
When Block B is placed on top of Block A, the total mass of the system becomes \( m_A + m_B \), where \( m_B \) is the mass of Block B. The problem states that the acceleration of the combined blocks drops to \( \frac{A}{5} \).

### Step 3: Write the equation for the combined system
The force acting on the combined system is still the same, and can be expressed as:
\[ F = (m_A + m_B) \cdot \frac{A}{5} \]

### Step 4: Set the equations equal to each other
Since the force remains constant, we can set the two expressions for force equal to each other:
\[ m_A \cdot A = (m_A + m_B) \cdot \frac{A}{5} \]

### Step 5: Simplify the equation
We can cancel \( A \) from both sides (since \( A \neq 0 \)):
\[ m_A = (m_A + m_B) \cdot \frac{1}{5} \]

### Step 6: Multiply both sides by 5
Multiplying both sides by 5 gives us:
\[ 5m_A = m_A + m_B \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate \( m_B \):
\[ 5m_A - m_A = m_B \]
\[ 4m_A = m_B \]

### Step 8: Find the ratio of the masses
Now, we can find the ratio of the mass of Block A to the mass of Block B:
\[ \frac{m_A}{m_B} = \frac{m_A}{4m_A} = \frac{1}{4} \]

### Conclusion
Thus, the ratio of the mass of Block A to the mass of Block B is:
\[ \frac{m_A}{m_B} = \frac{1}{4} \]

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