Question

Two particles having masses 4 g and 16 g respectively are moving with equal kinetic energies. The ratio of the magnitudes of their linear momentum is n : 2. The value of n will be ________ .

Hint:

When kinetic energy is constant for two particles, their linear momentum is directly proportional to the square root of their mass ($p \propto \sqrt{m}$). This is a direct consequence of the formula $p = \sqrt{2m(KE)}$.

Answer 1

Correct Answer: 1

The relationship between kinetic energy (KE) and linear momentum (p) is given by $KE = \frac{p^2}{2m}$.
This can be rearranged to express momentum as $p = \sqrt{2m(KE)}$.
Let the two particles be particle 1 and particle 2, with masses $m_1 = 4$ g and $m_2 = 16$ g.
We are given that their kinetic energies are equal: $KE_1 = KE_2$.
Let's find the ratio of their momenta, $p_1/p_2$:
$\frac{p_1}{p_2} = \frac{\sqrt{2m_1(KE_1)}}{\sqrt{2m_2(KE_2)}}$.
Since $KE_1 = KE_2$, the equation simplifies to:
$\frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}}$.
Substituting the given masses:
$\frac{p_1}{p_2} = \sqrt{\frac{4}{16}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$.
The ratio of their linear momenta is 1 : 2.
The problem states that the ratio is n : 2.
By comparing the two ratios, we find that n = 1.