Question

Two objects of masses 1 kg and 3 kg have equal momentum. What is the ratio of their kinetic energies ?

• A. \(3 : 1\)
• B. \(9 : 1\)
• C. \(1 : 1\)
• D. \(1 : 2\)

Hint:

Useful relationship: Kinetic Energy \(KE = \frac{p^2}{2m}\), where \(p\) is momentum and \(m\) is mass. Given: \(m_1 = 1\) kg, \(m_2 = 3\) kg, and \(p_1 = p_2 = p\) (equal momentum). Ratio of kinetic energies: \[ \frac{KE_1}{KE_2} = \frac{p^2/(2m_1)}{p^2/(2m_2)} \] Since \(p^2\) and 2 are common, they cancel out: \[ \frac{KE_1}{KE_2} = \frac{1/m_1}{1/m_2} = \frac{m_2}{m_1} \] Substitute the masses: \[ \frac{KE_1}{KE_2} = \frac{3 \text{ kg}}{1 \text{ kg}} = \frac{3}{1} \] So, the ratio is \(3:1\). This means if momentum is the same, the object with smaller mass has higher kinetic energy.

Answer 1

The Correct Option is A

Concept: Momentum (\(p\)) of an object is given by \(p = mv\), where \(m\) is mass and \(v\) is velocity. Kinetic Energy (KE) of an object is given by \(KE = \frac{1}{2}mv^2\). We can also express kinetic energy in terms of momentum: Since \(v = p/m\), substitute this into the KE formula: \(KE = \frac{1}{2}m\left(\frac{p}{m}\right)^2 = \frac{1}{2}m\frac{p^2}{m^2} = \frac{p^2}{2m}\). So, \(KE = \frac{p^2}{2m}\). Step 1: Define the given information for the two objects Let the first object have mass \(m_1 = 1 \text{ kg}\), momentum \(p_1\), and kinetic energy \(KE_1\). Let the second object have mass \(m_2 = 3 \text{ kg}\), momentum \(p_2\), and kinetic energy \(KE_2\). Given: The objects have equal momentum. So, \(p_1 = p_2\). Let this common momentum be \(p\). Step 2: Write the kinetic energy for each object using the formula \(KE = \frac{p^2}{2m}\) For the first object: \[ KE_1 = \frac{p_1^2}{2m_1} = \frac{p^2}{2(1)} = \frac{p^2}{2} \] For the second object: \[ KE_2 = \frac{p_2^2}{2m_2} = \frac{p^2}{2(3)} = \frac{p^2}{6} \] Step 3: Find the ratio of their kinetic energies (\(KE_1 : KE_2\)) We need to find \(\frac{KE_1}{KE_2}\). \[ \frac{KE_1}{KE_2} = \frac{\frac{p^2}{2}}{\frac{p^2}{6}} \] To divide these fractions, multiply the numerator by the reciprocal of the denominator: \[ \frac{KE_1}{KE_2} = \frac{p^2}{2} \times \frac{6}{p^2} \] The \(p^2\) terms cancel out: \[ \frac{KE_1}{KE_2} = \frac{6}{2} = \frac{3}{1} \] So, the ratio of their kinetic energies \(KE_1 : KE_2\) is \(3:1\). This matches option (1).