Question

Three bodies A, B and C have equal kinetic energies and their masses are 400 g, 1.2 kg and 1.6 kg respectively. The ratio of their linear momenta is :

• A. \(1 : \sqrt{3} : 2\)
• B. \(1 : \sqrt{3} : \sqrt{2}\)
• C. \(\sqrt{2} : \sqrt{3} : 1\)
• D. \(\sqrt{3} : \sqrt{2} : 1\)

Answer 1

The Correct Option is A

To determine the ratio of linear momenta for three bodies possessing equal kinetic energies, we commence by establishing the link between kinetic energy and momentum.

The formula for a body's kinetic energy (\(KE\)) is:

\(KE = \frac{1}{2}mv^2\)

where \(m\) denotes the mass and \(v\) denotes the velocity.

The linear momentum (\(p\)) of a body is defined as:

\(p = mv\)

Given that bodies A, B, and C have identical kinetic energies, the following equation holds for each:

\(\frac{1}{2} m_A v_A^2 = \frac{1}{2} m_B v_B^2 = \frac{1}{2} m_C v_C^2\)

Equating the kinetic energies, we derive:

\(m_A v_A^2 = m_B v_B^2 = m_C v_C^2\)

We can express each body's momentum in terms of its velocity:

\(v_A = \sqrt{\frac{2 \times KE}{m_A}}\), \(v_B = \sqrt{\frac{2 \times KE}{m_B}}\), \(v_C = \sqrt{\frac{2 \times KE}{m_C}}\)

Substituting these into the momentum formula yields:

\(p_A = m_A \times v_A = m_A \times \sqrt{\frac{2 \times KE}{m_A}}\), \(p_B = m_B \times v_B = m_B \times \sqrt{\frac{2 \times KE}{m_B}}\), \(p_C = m_C \times v_C = m_C \times \sqrt{\frac{2 \times KE}{m_C}}\)

Simplification of these expressions results in:

\(p_A = \sqrt{2 \times KE \times m_A}\), \(p_B = \sqrt{2 \times KE \times m_B}\), \(p_C = \sqrt{2 \times KE \times m_C}\)

The provided masses are \(m_A = 400\) g, \(m_B = 1.2\) kg, and \(m_C = 1.6\) kg. Converting \(m_A\) to kilograms yields \(m_A = 0.4\) kg.

Utilizing these masses to compute the momentum ratio:

Momentum ratio calculations:

\(\frac{p_A}{p_B} = \frac{\sqrt{0.4}}{\sqrt{1.2}} = \frac{\sqrt{1}}{\sqrt{3}} = \frac{1}{\sqrt{3}}\)

\(\frac{p_B}{p_C} = \frac{\sqrt{1.2}}{\sqrt{1.6}} = \frac{\sqrt{3}}{\sqrt{4}} = \frac{\sqrt{3}}{2}\)

Combining these ratios yields the complete momentum relationship:

\(p_A : p_B : p_C = 1 : \sqrt{3} : 2\)

Therefore, the correct ratio is \(1 : \sqrt{3} : 2\).