Question

A 4 kg mass and a 1 kg mass are moving with equal energies. The ratio of the magnitude of their linear momenta is

• A. 1 : 2
• B. 2 : 1
• C. 1 : 1
• D. 4 : 1

Answer 1

The Correct Option is B

KE₁ = KE₂ 
Using the formula for kinetic energy: 
\(\frac {1}{2}\) . m₁ . v₁² = \(\frac {1}{2}\) . m₂ . v₂² 
Since the energies are equal, we have: 
m₁ . v₁² = m₂ . v₂² 
Rearranging the equation: 
\((\frac {v_1}{v_2})^2\) = \(\frac {m_2}{m_1}\)
Taking the square root of both sides: 
\(\frac {v_1}{v_2}\)= \(\sqrt {\frac {m_2}{m_1}}\)
Now we can find the ratio of the magnitudes of their linear momenta: 
\(\frac {p_1}{p_2}\)= \(\frac {m_1.v_1}{m_2.v_2}\)
Substituting \(\frac {v_1}{v_2}\) = \(\sqrt{m_2 / m_1}\): 
\(\frac {p_1}{p_2}\) = \(\frac {m_1.v_1}{m_2.v_2}\)) 
\(\frac {p_1}{p_2}\)= \(\frac {m_1.v_1}{m_2.v_2}\) . \(\sqrt {\frac {m_1}{m_2}}\)
Canceling out v₁ and v₂: 
\(\frac {p_1}{p_2}\) = \(\frac {m_1/m_2}{\sqrt {m_1/m_2}}\) = \(\sqrt{m_2 / m_1}\)
Given that m₁ = 4 kg and m₂ = 1 kg: 
\(\frac {p_1}{p_2}\) = \(\sqrt{\frac {1} {4}}\) 
\(\frac {p_1}{p_2}\) = \(\frac {1}{2}\) 
Therefore, the ratio of the magnitudes of their linear momenta is (B) 2 : 1.