Two bodies of mass 4 g and 25 g are moving with equal kinetic energies. The ratio of the magnitude of their linear momentum is:
3:5
- 2:5
5:4
4:5
The Correct Option is B
Approach Solution - 1
To determine the ratio of the magnitudes of linear momentum for two bodies with equal kinetic energies, we need to understand the relationship between kinetic energy, mass, and momentum.
Concepts Used:
- Kinetic energy of an object is given by the formula: \(KE = \frac{1}{2} mv^2\), where \(m\) is the mass and \(v\) is the velocity of the object.
- Linear momentum of an object is given by: \(p = mv\).
Given:
- Mass of body 1, \(m_1 = 4\) g
- Mass of body 2, \(m_2 = 25\) g
- Kinetic energies are equal: \(KE_1 = KE_2\)
Step-by-step Solution:
1. Write the expression for kinetic energy for each body:
- For body 1: \(KE_1 = \frac{1}{2} m_1 v_1^2\)
- For body 2: \(KE_2 = \frac{1}{2} m_2 v_2^2\)
2. Since the kinetic energies are equal, we equate the expressions:
\(\frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_2 v_2^2\)
Cancel the \(\frac{1}{2}\) factor and rearrange the equation:
\(m_1 v_1^2 = m_2 v_2^2\)
3. From this, express the velocities in terms of each other:
\(v_1^2 = \frac{m_2}{m_1} v_2^2\)
4. Solve for the ratio of velocities:
\(\frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} = \sqrt{\frac{25}{4}} = \frac{5}{2}\)
5. The ratio of the magnitudes of linear momentum \((p_1/p_2)\) can be expressed as:
\(\frac{p_1}{p_2} = \frac{m_1 v_1}{m_2 v_2}\)
Substitute the values and expression for \(\frac{v_1}{v_2}\):
\(\frac{p_1}{p_2} = \frac{4 \cdot \frac{5}{2}v_2}{25 \cdot v_2} = \frac{20}{50} = \frac{2}{5}\)
The ratio of their linear momentum is \(2:5\).
Conclusion:
The correct answer is 2:5. This matches the given correct answer in the options.
Approach Solution -2
For objects with equal kinetic energies \(\left(\frac{p_1^2}{2m_1} = \frac{p_2^2}{2m_2}\right)\), we have:
\[\frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}}\]
Substituting \(m_1 = 4 \, \text{g}\) and \(m_2 = 25 \, \text{g}\):
\[\frac{p_1}{p_2} = \sqrt{\frac{4}{25}} = \frac{2}{5}\]
Thus, the ratio of their momenta is 2 : 5.
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Concepts Used:
Kinetic energy
Kinetic energy of an object is the measure of the work it does as a result of its motion. Kinetic energy is the type of energy that an object or particle has as a result of its movement. When an object is subjected to a net force, it accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving object or particle defined by both its mass and its velocity. Any combination of motions is possible, including translation (moving along a route from one spot to another), rotation around an axis, vibration, and any combination of motions.
