Understanding Kinetic Energy and Momentum

Kinetic energy ($\text{KE}$) is the energy a body possesses due to its motion. Momentum ($p$) is a measure of mass in motion. Both are fundamental concepts in physics and are related to the mass ($m$) and velocity ($v$) of a body.

The standard formulas are:

• Kinetic Energy: $\text{KE} = \frac{1}{2}mv^2$
• Momentum: $p = mv$

Relating Kinetic Energy, Momentum, and Mass

We can express kinetic energy in terms of momentum and mass. From the momentum formula, we can write velocity $v$ as $v = \frac{p}{m}$.

Substituting this expression for $v$ into the kinetic energy formula:

$\text{KE} = \frac{1}{2}m\left(\frac{p}{m}\right)^2$
$\text{KE} = \frac{1}{2}m\frac{p^2}{m^2}$
$\text{KE} = \frac{p^2}{2m}$

This formula $\left(\text{KE} = \frac{p^2}{2m}\right)$ shows how kinetic energy, momentum, and mass are related.

Analyzing the Change in Kinetic Energy

The question states that the momentum of the body remains unchanged, and the mass of the body is doubled.

Let the initial momentum, mass, and kinetic energy be $p_1$, $m_1$, and $\text{KE}_1$ respectively.

So, $\text{KE}_1 = \frac{p_1^2}{2m_1}$.

Now, let the new momentum, mass, and kinetic energy be $p_2$, $m_2$, and $\text{KE}_2$.

According to the question:

• Momentum remains unchanged: $p_2 = p_1$
• Mass is doubled: $m_2 = 2m_1$

Now, we calculate the new kinetic energy $\text{KE}_2$ using the formula $\text{KE} = \frac{p^2}{2m}$ with the new values $p_2$ and $m_2$:

$\text{KE}_2 = \frac{p_2^2}{2m_2}$
Substitute $p_2 = p_1$ and $m_2 = 2m_1$:
$\text{KE}_2 = \frac{(p_1)^2}{2(2m_1)}$
$\text{KE}_2 = \frac{p_1^2}{4m_1}$

Comparing Initial and Final Kinetic Energy

We have the initial kinetic energy $\text{KE}_1 = \frac{p_1^2}{2m_1}$ and the new kinetic energy $\text{KE}_2 = \frac{p_1^2}{4m_1}$.

We can write $\text{KE}_2$ in terms of $\text{KE}_1$:

$\text{KE}_2 = \frac{1}{2} \times \frac{p_1^2}{2m_1}$
Since $\text{KE}_1 = \frac{p_1^2}{2m_1}$, we get:
$\text{KE}_2 = \frac{1}{2} \text{KE}_1$

This shows that the new kinetic energy is half of the initial kinetic energy.

Conclusion

Keeping the momentum unchanged, if the mass of a body is doubled, its kinetic energy becomes half.