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Chapter 4: Problem 452

Two bodies of masses \(m_{1}\) and \(m_{2}\) have equal kinetic energies. If \(P_{1}\) and \(P_{2}\) are their respective momentum, what is ratio of \(\mathrm{P}_{2}: \mathrm{P}_{1}\) ? (A) \(\mathrm{m}_{1}: \mathrm{m}_{2}\) (B) \(\sqrt{\mathrm{m}}_{2} / \sqrt{\mathrm{m}_{1}}\) (C) \(\sqrt{m_{1}}: \sqrt{m_{2}}\) (D) \(\mathrm{m}_{1}^{2}: \mathrm{m}_{2}^{2}\)

Short Answer

Expert verified
The ratio of their momenta P2:P1 is \(\sqrt{m_{2}}: \sqrt{m_{1}}\). Therefore, the correct answer is (C) \(\sqrt{m_{1}}: \sqrt{m_{2}}\).

Step by step solution

01

Kinetic Energy formula:

The formula for kinetic energy (K) is given by: \[K = \frac{1}{2}mv^2\] where m is the mass of the body and v is its velocity.
02

Momentum formula:

The formula for momentum (P) is given by: \[P = m \cdot v\] where m is the mass of the body and v is its velocity.
03

Express the equal kinetic energies:

Since the kinetic energies of both bodies are equal, we can write: \[\frac{1}{2}m_1v_1^2 = \frac{1}{2}m_2v_2^2\]
04

Simplify the equation and solve for ratio of velocities:

We can cancel out the fractions and get the ratio of the velocities squared: \[m_1v_1^2 = m_2v_2^2\] \[\frac{v_{1}^{2}}{v_{2}^{2}} = \frac{m_{2}}{m_{1}}\]
05

Express the momenta:

We can write the momenta for both bodies using the momentum formula: \[P_1 = m_1v_1\] \[P_2 = m_2v_2\]
06

Find the ratio P2 : P1:

Divide the momentum formulas: \[\frac{P_2}{P_1} = \frac{m_2v_2}{m_1v_1}\]
07

Use the ratio of velocities squared from step 2:

Replace the ratio of velocities squared in the ratio of momenta: \[\frac{P_2}{P_1} = \frac{m_2}{m_1} \cdot \sqrt{\frac{v_{1}^{2}}{v_{2}^{2}}}\] Using the result from step 2, we get: \[\frac{P_2}{P_1} = \frac{m_2}{m_1} \cdot \sqrt{\frac{m_{2}}{m_{1}}}\]
08

Simplify the expression:

Now simplify the expression to get the final ratio: \[\frac{P_2}{P_1} = \sqrt{\frac{m_{2}^2}{m_{1}^2}}\] So, the ratio of their momenta P2:P1 is: \[\frac{P_2}{P_1} = \sqrt{m_{2}}: \sqrt{m_{1}}\] Therefore, the correct answer is (C) \(\sqrt{m_{1}}: \sqrt{m_{2}}\).

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