Question

Three bodies $P$, $Q$, and $R$ have masses $m\text{ kg}$, $2m\text{ kg}$, and $3m\text{ kg}$ respectively. If all the bodies have equal kinetic energy, then the greater momentum will be for body/bodies

• A. $Q$
• B. $R$
• C. $P$ and $Q$
• D. $P$

Hint:

A useful conceptual shortcut to remember for physics exams is: under equal kinetic energy constraints, the heavier body always carries more momentum ($p \propto \sqrt{m}$). Conversely, under equal momentum constraints, the lighter body always carries more kinetic energy ($E_k \propto \frac{1}{m}$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given three bodies with different masses scaled by a factor $m$. The problem states that all three bodies possess identical kinetic energies ($E_k$). We need to identify which body develops the maximum linear momentum ($p$).

Step 2: Key Formula or Approach:
The fundamental formula relating linear momentum ($p = mv$) and kinetic energy ($E_k = \frac{1}{2}mv^2$) is expressed as: $$E_k = \frac{p^2}{2m} \implies p = \sqrt{2mE_k}$$ Since the problem states that the kinetic energy $E_k$ is a constant value across all three systems, the linear momentum is directly proportional to the square root of the mass of the body: $$p \propto \sqrt{m}$$

Step 3: Detailed Explanation:
Let's analyze the proportional relationship established by our formula: $$p_P \propto \sqrt{m}$$ $$p_Q \propto \sqrt{2m}$$ $$p_R \propto \sqrt{3m}$$ Comparing the masses of the three bodies: $$m < 2m < 3m \implies m_P < m_Q < m_R$$ Taking the square root preserving inequalities: $$\sqrt{m_P} < \sqrt{m_Q} < \sqrt{m_R} \implies p_P < p_Q < p_R$$ Since body $R$ has the largest mass ($3m$), it inherently possesses the greatest linear momentum under equal kinetic energy criteria.

Step 4: Final Answer:
The body with the greater momentum is $R$, matching option (B).