If a lighter body (mass `M_(1)` and velocity `V_(1)` ) and a heavier body (mass `M_(2)` and velocity `V_(2)`) have the same kinetic energy, then-

Options

A. `M_(2)V_(2) lt M_(1)V_(1)`
B. `M_(2)V_(2)=M_(1)V_(1)`
C. `M_(2)V_(1)=M_(1)V_(2)`
D. `M_(2)V_(2) gt M_(1)V_(1)`

Answer

To solve the problem, we need to analyze the kinetic energies of the two bodies and set them equal to each other since they are given to have the same kinetic energy.

### Step 1: Write the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \(m\) is the mass of the object and \(v\) is its velocity.

### Step 2: Write the kinetic energy equations for both bodies
For the lighter body (mass \(M_1\) and velocity \(V_1\)):
\[ KE_1 = \frac{1}{2} M_1 V_1^2 \]

For the heavier body (mass \(M_2\) and velocity \(V_2\)):
\[ KE_2 = \frac{1}{2} M_2 V_2^2 \]

### Step 3: Set the kinetic energies equal to each other
Since both bodies have the same kinetic energy, we can set the two equations equal to each other:
\[ \frac{1}{2} M_1 V_1^2 = \frac{1}{2} M_2 V_2^2 \]

### Step 4: Simplify the equation
We can cancel \(\frac{1}{2}\) from both sides:
\[ M_1 V_1^2 = M_2 V_2^2 \]

### Step 5: Rearranging the equation
We can rearrange the equation to express the relationship between the masses and velocities:
\[ \frac{V_1^2}{V_2^2} = \frac{M_2}{M_1} \]

### Step 6: Take the square root
Taking the square root of both sides gives us:
\[ \frac{V_1}{V_2} = \sqrt{\frac{M_2}{M_1}} \]

### Conclusion
Thus, the relationship between the velocities of the two bodies is given by:
\[ V_1 = V_2 \sqrt{\frac{M_2}{M_1}} \]
This indicates that the velocity of the lighter body is proportional to the square root of the ratio of the heavier body’s mass to the lighter body’s mass.

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