Question

Four particles A, B, C, D of mass $\frac{m}{2}$, $m$, $2m$, $4m$ have the same momentum, respectively. The particle with maximum kinetic energy is:

• A. D
• B. C
• C. A
• D. B

Answer 1

The Correct Option is C

The kinetic energy is given by:

\[ KE = \frac{p^2}{2m} \]

Since all particles have the same momentum, the kinetic energy is inversely proportional to their mass:

\[ KE \propto \frac{1}{m} \]

Thus, the particle with the smallest mass will have the maximum kinetic energy.

Among the given particles:

\[ m_A = \frac{m}{2}, \quad m_B = m, \quad m_C = 2m, \quad m_D = 4m \]

Hence, \(\frac{m}{2}\) (particle A) has the maximum kinetic energy.

Answer 2

To determine which particle has the maximum kinetic energy, we need to analyze the kinetic energy formula in terms of momentum. The kinetic energy \(K\) of a particle can be expressed as:

\(K = \frac{p^2}{2m}\)

where \(p\) is the momentum of the particle and \(m\) is its mass.

Given that all particles have the same momentum, we can denote this common momentum by \(p\).

Now, let's calculate the kinetic energy for each particle:

1. For particle A, with mass \(\frac{m}{2}\):

\(K_A = \frac{p^2}{2 \times \frac{m}{2}} = \frac{p^2}{m}\)

1. For particle B, with mass \(m\):

\(K_B = \frac{p^2}{2 \times m} = \frac{p^2}{2m}\)

1. For particle C, with mass \(2m\):

\(K_C = \frac{p^2}{2 \times 2m} = \frac{p^2}{4m}\)

1. For particle D, with mass \(4m\):

\(K_D = \frac{p^2}{2 \times 4m} = \frac{p^2}{8m}\)

Comparing these expressions, we see:

• \(K_A = \frac{p^2}{m}\)
• \(K_B = \frac{p^2}{2m}\)
• \(K_C = \frac{p^2}{4m}\)
• \(K_D = \frac{p^2}{8m}\)

Since \(\frac{p^2}{m}\) (Particle A) is the largest value among all, particle A has the maximum kinetic energy.

Thus, the correct is A.