Question

Two bodies of mass ${m_A}$ and ${m_B}$ have equal kinetic energy. The ratio of their momenta is

A. $\sqrt {{m_A}} :\sqrt {{m_B}}$

B. ${m_A}^2:{m_B}^2$

C. ${m_A}:{m_B}$

D. ${m_B}:{m_A}$

Answer 1

When the two bodies of different masses have the same kinetic energy, by equating the formula of the kinetic energy of two bodies, we can get the mass and velocity relation of the two bodies. Then, replacing the mass and velocity terms with momentum, we get the momentum and velocity relation of two bodies. Finally, substitute the velocity with mass obtained by relating the kinetic energy, thus we can derive the ratio of the momenta in terms of the mass of the two bodies.

Complete step by step answer:

The kinetic energy can be expressed as,

$KE = \dfrac{1}{2} \times m \times {v^2}$

Where ,$m$ is the mass of a body and $v$ is the velocity of the body.

Since the kinetic energy of two bodies of mass ${m_A}$ and ${m_B}$ are equal. Then,

$\dfrac{1}{2} \times {m_A} \times {v_A}^2 = \dfrac{1}{2} \times {m_B} \times {v_B}^2\;..........\left( 1 \right)$

Equation (1) can be simplified as,

${m_A} \times {v_A}^2 = {m_B} \times {v_B}^2\;...........\left( 2 \right)$

Momentum can be calculated as mass multiplied with velocity of the body and it is expressed as,

$p = m \times v\;.........\left( 3 \right)$

Where, $m$ is the mass of the object and $v$ is the velocity of the object.

From Equation (2)

$\dfrac{{m_A}^2 \times {v_A}^2}{m_A} = \dfrac{{m_B}^2 \times {v_B}^2}{m_B}..........(4)$

Now substitute the equation (3) in equation (4)

$\dfrac{{p_A}^2}{v_A} = \dfrac{{p_B}^2}{v_B}$

Above equation can be arranged as

$\dfrac{{p_A}^2}{{p_B}^2} = \dfrac{m_A}{m_B}$

By taking square root on both sides

$\dfrac{{{p_A}}}{{{p_B}}} = \sqrt {\dfrac{{{m_A}}}{{{m_B}}}} \;...........\left( 5 \right)$

Thus, the two bodies of mass ${m_A}$ and ${m_B}$ which has equal kinetic energy have the ratio of momenta is $\sqrt {{m_A}} :\sqrt {{m_B}}$.

Hence, the option (A) is correct.

Note:
The constant of proportionality is determined by experimental results in most cases. It gives us an idea about the extent to which a quantity increases or decreases if we keep other variables with unit value. It is also known as the ratio between two proportional quantities.