Question

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

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

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.

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.

Express the equal kinetic energies:

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

Simplify the equation and solve for ratio of velocities:

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

Express the momenta:

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

Find the ratio P2 : P1:

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

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: \[\frac{P_2}{P_1} = \frac{m_2}{m_1} \cdot \sqrt{\frac{m_{2}}{m_{1}}}\]

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) \(\sqrt{m_{1}}: \sqrt{m_{2}}\).

Key concepts

Kinetic Energy

Kinetic energy is a key concept in physics that measures the energy that an object possesses due to its motion. It is an expression of how much work an object can do because of its speed. The formula for kinetic energy is:

• \(K = \frac{1}{2}mv^2\)

where \(m\) stands for the mass of the object and \(v\) for its velocity. This relationship tells us that kinetic energy is directly proportional to the mass and the square of the velocity. In simpler terms:

• Doubling the mass will double the kinetic energy, assuming velocity stays constant.
• Doubling the velocity will fourfold increase the kinetic energy because velocity is squared in the formula.

Understanding this formula is crucial as it helps to explain why heavier or faster objects tend to have higher kinetic energy.

Mass

Mass is a fundamental property of an object that represents the quantity of matter it contains. Different from weight, mass is invariant—meaning it doesn't change based on location. When we talk about mass in physics, especially in relation to momentum and kinetic energy, it plays a significant role in determining how much energy an object has in motion or how it behaves under various forces.Knowing how mass interacts with other quantities:

• It influences the momentum of an object (\(P = m \cdot v\)). More mass means more momentum if velocity remains constant.
• Contributes directly to kinetic energy (\(K = \frac{1}{2}mv^2\)). A larger mass leads to greater kinetic energy at the same speed.

Mass serves as a cornerstone in both kinetic energy and momentum equations, highlighting its critical role in motion-related calculations.

Velocity

Velocity is a vector quantity that not only refers to the speed of an object but also specifies its direction of motion. It is a fundamental aspect of many physics equations, notably those concerning kinetic energy and momentum.In terms of momentum and kinetic energy:

• Velocity is squared in the kinetic energy formula \(K = \frac{1}{2}mv^2\), which shows its exponential influence on the energy of a moving object.
• Velocity is used directly in the momentum equation \(P = m \cdot v\), impacting the momentum linearly. Thus, an increase in velocity results in a proportional increase in momentum if mass is steady.

In order to solve physics problems where kinetic energy or momentum is involved, it's crucial to understand both the speed and the direction depicted by velocity.

Formula

Formulas in physics, like the ones for kinetic energy and momentum, serve as mathematical models for understanding the physical world. They offer a streamlined way to calculate complex phenomena by connecting measurable quantities, such as mass and velocity.To elaborate:

• Kinetic energy formula \(K = \frac{1}{2}mv^2\) highlights the relationship between movement and energy. It shows how mass and speed of an object contribute to its energy.
• Momentum formula \(P = m \cdot v\) links the mass and velocity of an object to describe its motion's impact.
• The problems involving these formulas require a clear understanding of their derivations and implications—for instance, knowing how to isolate variables or substitute values useful in solving real-world problems.

Understanding and applying these formulas allow students to calculate energy and momentum, equipping them with the tools needed to tackle various physics challenges.