A constant force actson a body of mass m, at rest and produces a velocity 'v' over a displacement '`s_1`'. The same force acts on another body of mass `m_2` at rest and produces the same velocity. 'v' over a displacement '`s_2`'. The ratio of the displacements is

Options

A. `m_1 : m_2`
B. `m_2 : m_1`
C. `sqrt(m_1) : sqrt(m_2)`
D. `sqrt(m_2) : sqrt(m_1)`

Answer

To solve the problem, we need to analyze the motion of two bodies under the influence of the same constant force. Here's the step-by-step solution:

### Step 1: Understand the given information
We have two bodies:
- Body 1 with mass \( m_1 = m \) (mass of the first body)
- Body 2 with mass \( m_2 \)

Both bodies start from rest and reach the same final velocity \( v \) over different displacements \( s_1 \) and \( s_2 \) respectively.

### Step 2: Apply the third equation of motion
The third equation of motion relates initial velocity, final velocity, acceleration, and displacement:
\[ v^2 = u^2 + 2as \]
Since both bodies start from rest, the initial velocity \( u = 0 \). Thus, the equation simplifies to:
\[ v^2 = 2as \]
Where \( a \) is the acceleration and \( s \) is the displacement.

### Step 3: Express acceleration in terms of force and mass
From Newton's second law, we know that:
\[ F = ma \]
Thus, the acceleration \( a \) can be expressed as:
\[ a = \frac{F}{m} \]

### Step 4: Substitute acceleration into the equation
Substituting \( a \) into the equation \( v^2 = 2as \):
\[ v^2 = 2 \left(\frac{F}{m}\right) s \]
Rearranging gives:
\[ s = \frac{mv^2}{2F} \]

### Step 5: Write the equations for both bodies
For body 1 (mass \( m_1 = m \)):
\[ s_1 = \frac{mv^2}{2F} \]

For body 2 (mass \( m_2 \)):
\[ s_2 = \frac{m_2v^2}{2F} \]

### Step 6: Find the ratio of displacements
Now we can find the ratio of the displacements \( \frac{s_1}{s_2} \):
\[ \frac{s_1}{s_2} = \frac{\frac{mv^2}{2F}}{\frac{m_2v^2}{2F}} = \frac{m}{m_2} \]

Thus, the ratio of the displacements is:
\[ \frac{s_1}{s_2} = \frac{m}{m_2} \]

### Final Answer
The ratio of the displacements \( s_1 \) and \( s_2 \) is \( \frac{m}{m_2} \).

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