Problem 454 - JEE Physics

Question

Two bodies of masses \(\mathrm{m}\) and \(3 \mathrm{~m}\) have same momentum. their respective kinetic energies \(E_{1}\) and \(E_{2}\) are in the ratio..... (A) \(1: 3\) (B) \(3: 1\) (C) \(1: 3\) (D) \(1: 6\)

Step-by-Step Solution

Answer

The ratio of the kinetic energies \(E_1\) and \(E_2\) is \(3:1\).

Step 1: Momenta of the objects

We are given that the two objects have the same momentum. So let's write the momenta for both objects: Object 1 (mass = m): Momenta = m * v1 Object 2 (mass = 3m): Momenta = 3m * v2 According to the problem, the momenta are equal, so: \(m * v1 = 3m * v2\)

Step 2: Solving for one of the speeds

Dividing both sides of the equation by m gives: \(v1 = 3 * v2\) Now let's solve for either v1 or v2. We choose v2 for simplicity: \(v2 = \frac{v1}{3}\)

Step 3: Writing expressions for kinetic energies

Now let's write the kinetic energy expressions for each object: Object 1 (mass = m): \(E_1 = \frac{1}{2} * m * v1^2\) Object 2 (mass = 3m): \(E_2 = \frac{1}{2} * 3m * v2^2\)

Step 4: Substituting expression for v2 and simplifying

Now, let's substitute the expression for v2 from Step 2 into the expression for E2: \(E_2 = \frac{1}{2} * 3m * \left(\frac{v1}{3}\right)^2\) And simplifying E2: \(E_2 = \frac{1}{2} * 3m * \frac{v1^2}{9}\) \(E_2 = \frac{1}{6} * m * v1^2\)

Step 5: Finding the ratio of kinetic energies

To find the ratio of the kinetic energies, we will divide \(E_1\) by \(E_2\): \(\frac{E_1}{E_2} = \frac{\frac{1}{2} * m * v1^2}{\frac{1}{6} * m * v1^2}\) Cancelling the terms \(m * v1^2\): \(\frac{E_1}{E_2} = \frac{\frac{1}{2}}{\frac{1}{6}}\) Now, dividing the fractions: \(\frac{E_1}{E_2} = \frac{1}{2} * \frac{6}{1}\) \(\frac{E_1}{E_2} = 3\) Thus, \(E_1\) and \(E_2\) are in the ratio \(3:1\), which corresponds to option (B).

Key Concepts

Momentum Conservation

Momentum, a fundamental concept in physics, is defined as the product of an object's mass and velocity. When two bodies have the same momentum, it means that their momentum magnitudes are equal, even if their individual masses and velocities differ. In the original exercise, two objects with masses of \( m \) and \( 3m \) have the same momentum.

• For the first object (with mass \( m \)), momentum is expressed as \( m \times v_1 \).
• For the second object (with mass \( 3m \)), it is \( 3m \times v_2 \).

Given their momenta are equal: \( m \times v_1 = 3m \times v_2 \), which simplifies to \( v_1 = 3v_2 \).
This relationship tells us that despite the differing masses, the product of mass and velocity remains constant when momentum is conserved. This characteristic is especially important in collision problems in physics, where understanding these relationships helps predict the motion of objects after a collision.

Mass and Velocity Relationship

The relationship between mass and velocity is crucial, particularly when discussing objects with equal momentum. When two objects have the same momentum, a change in one parameter (mass or velocity) must be balanced by a change in the other. In our scenario, since the object with larger mass (\(3m\)) must have a correspondingly smaller velocity (\(v_2 = \frac{v_1}{3}\)).
This inverse relationship ensures that the momentum remains equal despite disparities in mass. So, if an object's mass is tripled, its velocity must be one-third that of a lighter object with the same momentum.

• This concept is why heavier vehicles, like trucks, require more energy to reach high speeds compared to lighter vehicles. The same momentum requires balancing their larger mass with proportionately less velocity.
• Understanding this helps solve numerous real-world physics problems, providing a clear picture of how mass and velocity interact.

Ratio of Kinetic Energies

Kinetic energy is the energy an object possesses due to its motion, calculated as \( \text{KE} = \frac{1}{2} \cdot m \cdot v^2 \). In scenarios with equal momentum but differing masses, their kinetic energies can vary significantly. Let's explore this using our two objects:

• For the first object (mass \( m \)), kinetic energy is \( E_1 = \frac{1}{2} \cdot m \cdot v_1^2 \).
• For the second object (mass \( 3m \)), kinetic energy becomes \( E_2 = \frac{1}{2} \cdot 3m \cdot v_2^2 \).

Substituting \( v_2 = \frac{v_1}{3} \) into the equation for \( E_2 \), we find that \( E_2 = \frac{1}{6} \cdot m \cdot v_1^2 \). This leads us to conclude: \( \frac{E_1}{E_2} = 3 \), or the energies are in a ratio of 3:1.

• This outcome stems from the velocity's squared effect in kinetic energy formula, causing greater disparity for larger masses.
• Recognizing these energy differences when momentum is conserved aids in understanding energy distribution in physical systems, such as during collisions.