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

Answer: (B)

2 : 1

Answer 2

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.