Question

Two bodies having the same linear momentum have a ratio of kinetic energy as 16:9. Find the ratio of masses of these bodies.

• A. \(\frac{9}{16}\)
• B. \(\frac{4}{3}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{16}{9}\)

Hint:

When two bodies have the same linear momentum but different kinetic energies, the body with the greater kinetic energy must have a smaller mass. This relationship arises because kinetic energy depends on both mass and the square of velocity, whereas momentum depends linearly on both.

Answer 1

The Correct Option is A

To solve the given problem, we need to understand the relationship between linear momentum, kinetic energy, and mass. Let's go through the steps:

1. We are given that two bodies have the same linear momentum. Let the linear momentum of each body be \( p \). Thus, for both bodies, the momentum is \( p = m_1 v_1 = m_2 v_2 \), where \( m_1, v_1 \) are the mass and velocity of the first body, and \( m_2, v_2 \) are those of the second body.
2. The kinetic energy (\( KE \)) of a body is given by the formula:

\(KE = \frac{1}{2} m v^2\)

1. Now, substituting \( v \) from the momentum equation \( p = mv \) gives:

\(v = \frac{p}{m}\)

1. Therefore, the kinetic energy can be expressed as:

\(KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m}\)

1. According to the problem, the ratio of kinetic energies of the two bodies is given as 16:9. Thus:

\(\frac{KE_1}{KE_2} = \frac{16}{9}\)

1. Using the kinetic energy formula, substitute for each body:

\(\frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{m_2}{m_1} = \frac{16}{9}\)

1. From the above equation, we can equate as:

\(\frac{m_2}{m_1} = \frac{16}{9}\)

1. Therefore, the ratio of masses \( \frac{m_1}{m_2} \) is the reciprocal:

\(\frac{m_1}{m_2} = \frac{9}{16}\)

Thus, the correct answer is that the ratio of the masses of these bodies is \(\frac{9}{16}\).