A mass `m_(1)` is moving on a plane surface along a straight path under the action of a force F. Another mass `m_(2)` is added on to `m_(1)` and the acceleration drops to 1/5 of the ear her value of the acceleration. Assuming that F remains constant, what is the ratio `m_(1) : m_(2)` ?

Options

A. `1//6`
B. `1//5`
C. `1//4`
D. `4//5`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the initial conditions
Let the initial mass be \( m_1 \) and the initial acceleration be \( a \). The force acting on the mass is given by Newton's second law:
\[ F = m_1 \cdot a \]

### Step 2: Understand the conditions after adding the second mass
When a second mass \( m_2 \) is added to the first mass, the new total mass becomes \( m_1 + m_2 \). The new acceleration, as per the problem, is \( \frac{1}{5} \) of the initial acceleration:
\[ a' = \frac{a}{5} \]

### Step 3: Write the equation for the new conditions
Since the force \( F \) remains constant, we can write the equation for the new situation:
\[ F = (m_1 + m_2) \cdot a' \]
Substituting \( a' \) into the equation gives:
\[ F = (m_1 + m_2) \cdot \frac{a}{5} \]

### Step 4: Set the two expressions for force equal to each other
Since both expressions represent the same force \( F \), we can equate them:
\[ m_1 \cdot a = (m_1 + m_2) \cdot \frac{a}{5} \]

### Step 5: Simplify the equation
To eliminate \( a \) from both sides (assuming \( a \neq 0 \)), we can divide both sides by \( a \):
\[ m_1 = (m_1 + m_2) \cdot \frac{1}{5} \]

### Step 6: Multiply through by 5 to eliminate the fraction
Multiplying both sides by 5 gives:
\[ 5m_1 = m_1 + m_2 \]

### Step 7: Rearrange the equation to find \( m_2 \)
Rearranging the equation gives:
\[ 5m_1 - m_1 = m_2 \]
\[ 4m_1 = m_2 \]

### Step 8: Find the ratio \( m_1 : m_2 \)
Now, we can express the ratio \( \frac{m_1}{m_2} \):
\[ \frac{m_1}{m_2} = \frac{m_1}{4m_1} = \frac{1}{4} \]

Thus, the ratio \( m_1 : m_2 \) is:
\[ m_1 : m_2 = 1 : 4 \]

### Final Answer
The ratio \( m_1 : m_2 \) is \( 1 : 4 \).

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