Exploring the Relationship between Mass, Kinetic Energy, and Velocity: A Comprehensive Guide

Understanding the relationship between mass, kinetic energy, and velocity is crucial in physics, particularly in the realm of mechanical systems and energy dynamics. In this article, we will delve into the calculations and concepts implied in the question 'Two bodies of masses 1 kg and 3 kg have equal kinetic energy; what is the ratio of their velocities?'

Understanding Kinetic Energy

The formula for kinetic energy (KE) is given by:

Formula: KE 1/2 mv^2

Here, m is the mass of the body, and v is its velocity.

Applying the Concept to the Given Problem

The given problem involves two bodies, one with a mass of 1 kg (body 1) and another with a mass of 3 kg (body 2). These bodies are said to have equal kinetic energies. We need to determine the ratio of their velocities. Let's break this down step by step:

Formulating the Equations

The kinetic energy of the first body (1 kg) is given by:

KE1 1/2 * 1 * v12 1/2 * v12

The kinetic energy of the second body (3 kg) is given by:

KE2 1/2 * 3 * v22 3/2 * v22

Setting the Kinetic Energies Equal

Given that the two bodies have equal kinetic energies, we set the above two equations equal to each other:

1/2 * v12 3/2 * v22

To simplify, we can cancel out the common factor of 1/2 from both sides of the equation:

v12 3 * v22

Solving for the Velocity Ratio

By taking the square root of both sides, we get:

v1 √3 * v2

Cancelling v2 from both sides, we obtain the ratio of their velocities:

v1 : v2 √3 : 1

Interpreting the Result

The result indicates that the velocity of the 1 kg mass is √3 times that of the 3 kg mass. This is a direct consequence of the inverse square relationship between kinetic energy and velocity for bodies with different masses but equal kinetic energy.

Applications and Insights

Understanding this concept is essential in various fields, including sports, engineering, and astronomy. For instance, in sports physics, this principle can be used to analyze the performance of athletes of different masses. In engineering, it helps in designing mechanisms that can handle different masses with the same energy input.

Conclusion

In conclusion, the relationship between mass, kinetic energy, and velocity is a fundamental concept in physics. The problem of two bodies with equal kinetic energies but different masses leads to a unique velocity ratio, which can be calculated using the kinetic energy formula. This ratio, √3 : 1, illustrates the inverse square relationship and provides valuable insights into the dynamics of physical systems.