Problem 24 - Master Resource Book in JEE Main Physics

Question

Two particles \(A_{1 s}\) and \(A_{2}\) of masses \(m_{1}, m_{2}\left(m_{1}>m_{2}\right)\) have the same de-broglie wavelength. Then [NCERT Exemplar] (a) their momenta are the same (b) their energies are the same (c) energy of \(A_{1}\) is less than the energy of \(A_{2}\) (d) energy of \(A_{1}\) is more than the energy of \(A_{2}\)

Step-by-Step Solution

Answer

(c) Energy of \(A_1\) is less than the energy of \(A_2\).

Step 1: Understanding De-broglie Wavelength

The de Broglie wavelength \( \lambda \) of a particle is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

Step 2: Comparing Wavelengths of Particles

We are given that both particles have the same de-broglie wavelength, implying that \( \frac{h}{p_1} = \frac{h}{p_2} \), where \( p_1 \) and \( p_2 \) are the momenta of particles \( A_1 \) and \( A_2 \) respectively. Hence, \( p_1 = p_2 \).

Step 3: Momentum vs. Mass and Velocity

Momentum \( p \) of a particle is also defined as \( p = mv \), where \( v \) is the velocity and \( m \) is the mass. Given that \( m_1 > m_2 \) and \( p_1 = p_2 \), the velocity \( v_1 \) of particle \( A_1 \) must be less than the velocity \( v_2 \) of particle \( A_2 \) to satisfy the equality.

Step 4: Energy Calculation for Particles

The kinetic energy \( E \) of a particle is given by \( E = \frac{1}{2}mv^2 \). Since \( v_1 < v_2 \) and \( m_1 > m_2 \), the kinetic energy \( E_1 = \frac{1}{2}m_1v_1^2 \) of particle \( A_1 \) will be less than the kinetic energy \( E_2 = \frac{1}{2}m_2v_2^2 \) of particle \( A_2 \) because the velocity term has a larger impact in this case.

Step 5: Concluding the Energy Comparison

As calculated, \( E_1 < E_2 \), meaning the energy of \( A_1 \) is less than the energy of \( A_2 \). Therefore, the correct answer is option (c).

Key Concepts

Momentum

Momentum is a fundamental concept in physics that describes the motion of a particle. It is calculated as the product of an object's mass and velocity, represented by the formula: \[ p = mv \] where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity. This formula teaches us that momentum depends directly on both how much stuff (mass) an object has and how fast it's moving (velocity).

A key point to remember is that even if two particles have different masses, they can have the same momentum if their velocities are adjusted accordingly. In the exercise above, although particle \( A_1 \) has a greater mass than \( A_2 \), the de Broglie wavelength for both particles is the same. This means they must have equal momenta, as evidenced by the same de Broglie wavelength equation \( \lambda = \frac{h}{p} \). This tells us that momentum is a critical parameter when comparing particles in motion, especially in quantum mechanics.

• Larger mass typically means slower velocity if momentum is constant.
• Equal momenta mean the balance between mass and velocity.

Kinetic Energy

Kinetic energy is the energy that a particle possesses due to its motion. It's defined by the formula: \[ E = \frac{1}{2}mv^2 \] where \( E \) is the kinetic energy, \( m \) is the mass, and \( v \) is the velocity. This formula highlights that kinetic energy depends on both the mass and the square of the velocity of the object. Notice that since velocity is squared, it has a much larger impact on the kinetic energy than the mass does.

In the given problem, even though particle \( A_1 \) has a greater mass, its kinetic energy is less than that of particle \( A_2 \). This happens because \( A_1 \) has a lower velocity to keep its momentum equal to \( A_2 \). The squaring of \( v \) in the formula means even a slight decrease in velocity results in a significant decrease in kinetic energy.

• Kinetic energy is influenced more by velocity changes than mass.
• Smaller velocities can lead to lower energies even for larger masses.

Mass and Velocity Relationship

The relationship between mass and velocity is essential to understanding both momentum and kinetic energy. As we got to see from the formula: \[ p = mv \] In scenarios where momentum is conserved or remains constant, the mass and velocity of a particle are inversely related. If one parameter increases, the other must decrease to maintain the same level of momentum. For example, if you make a particle heavier and still want the same momentum, it will move slower.

In our exercise, although \( m_1 > m_2 \), particle \( A_1 \) has a lower velocity than particle \( A_2 \) for their momenta to remain equal—because momentum \( p \) depends equally on both mass and velocity. This relationship helps explain why the kinetic energy can differ; a higher mass isn’t enough to make up for the reduced velocity in terms of energy because kinetic energy involves the square of velocity \( v^2 \), increasing its significance.

• Increasing mass leads to a decrease in velocity if momentum is constant.
• Velocity changes have a more pronounced effect on energy, considering the squared term in kinetic energy.