Problem 56 - Master Resource Book in JEE Main Physics

Question

Two masses of \(1 \mathrm{~g}\) and \(4 \mathrm{~g}\) are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is (a) \(4: 1\) (b) \(\sqrt{2}: 1\) (c) \(1: 2\) (d) \(1: 16\)

Step-by-Step Solution

Answer

The ratio of the magnitudes of their linear momenta is \(1:2\).

Step 1: Write the expression for kinetic energy

The kinetic energy (KE) of an object with mass \(m\) moving at velocity \(v\) is given by the formula \( KE = \frac{1}{2}mv^2 \). Since both masses have equal kinetic energy, we can set up an equation for each mass with this relation.

Step 2: Write the expression in terms of velocity

Let \( m_1 = 1 \, \text{g} \) and \( m_2 = 4 \, \text{g} \). We know \( KE_1 = KE_2 \). So, \( \frac{1}{2}m_1v_1^2 = \frac{1}{2}m_2v_2^2 \). This means \( m_1v_1^2 = m_2v_2^2 \). Substituting for \( m_1 \) and \( m_2 \), we get \( (1) v_1^2 = (4) v_2^2 \).

Step 3: Solve for velocity ratio

From \( v_1^2 = 4v_2^2 \), solve for the ratio of velocities: \( v_1 = 2v_2 \). Thus, velocity of the first mass is twice that of the second mass.

Step 4: Express momentum in terms of velocity

The momentum \( p \) of an object with mass \( m \) and velocity \( v \) is \( p = mv \). Therefore, the momentum of the two masses are \( p_1 = m_1v_1 \) and \( p_2 = m_2v_2 \).

Step 5: Calculate the ratio of momenta

The ratio of their momenta is \( \frac{p_1}{p_2} = \frac{m_1v_1}{m_2v_2} = \frac{1 \times (2v_2)}{4 \times v_2} = \frac{2}{4} = \frac{1}{2} \).

Step 6: Conclusion

The ratio of the magnitudes of their linear momenta is \( 1:2 \). Therefore, the correct answer is (c) \(1: 2\).

Key Concepts

linear momentum

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is given by the product of an object's mass and its velocity, expressed as \( p = mv \), where \( p \) represents momentum, \( m \) stands for mass, and \( v \) is the velocity of the object.
Understanding linear momentum is crucial because it tells us how difficult it is to stop a moving object. The greater the momentum, the harder it is to bring the object to a halt.

• For example, a truck moving at a certain velocity has more momentum than a bicycle moving at the same velocity because the truck's mass is much larger.
• Similarly, if two objects have the same mass, the one with the higher velocity will have more momentum.

This exercise involves comparing the linear momentum of two different masses, each having the same kinetic energy, to understand how their momenta differ. By setting up their kinetic energies to be equal, we explore how their velocities differ, which in turn affects their linear momentum. As demonstrated in the solution, even with equal kinetic energies, the ratio of their linear momenta is determined by their mass-to-velocity relationship.

velocity

Velocity refers to the speed of an object in a given direction. It is a vector quantity, meaning it has both magnitude and direction. In our exercise, understanding velocity is key to determining how two objects with the same kinetic energy but different masses compare.
Velocity is central to the problem because kinetic energy \( KE \) depends on the square of the velocity, given by the formula:
\[ KE = \frac{1}{2}mv^2 \]
Here are some important points about velocity:

• A slight change in velocity results in a more significant change in kinetic energy because of the squaring effect.
• When two objects have the same kinetic energy, a difference in mass directly influences their velocities.

In the solution, after equating the kinetic energies for the two masses, the relationship \( v_1 = 2v_2 \) was found. This means that the velocity of the first mass is twice that of the second mass, which highlights the inverse relationship between mass and velocity when kinetic energy remains constant. Larger masses require smaller velocities to maintain the same kinetic energy, and understanding this inverse relationship helps explain how the momenta of the objects are derived from their velocities.

mass ratio

The mass ratio is a simple comparison of the masses of two objects, expressed as a fraction. It tells us how one mass relates to another.
In the exercise, we have two objects with masses \( m_1 = 1 \, \text{g} \) and \( m_2 = 4 \, \text{g} \). The mass ratio can be expressed as \( \frac{m_1}{m_2} = \frac{1}{4} \).
Understanding the mass ratio helps us see its effect on other physical properties like velocity and momentum when energy limits are in play.

• For example, when two masses have identical kinetic energy, the mass ratio implies that the larger mass must move at a reduced velocity to keep the energy the same.
• The mass ratio simplifies analysis, showing how mass differences affect motion properties like velocity and momentum.

In the context of this exercise, the mass ratio \( \frac{1}{4} \) contributes to the final momentum ratio \( \frac{1}{2} \). This teaches us that while objects may have proportionally different masses, their kinetic energies and subsequent velocities align these masses in reconsidering their motion properties like momentum.