Exploding a Body at Rest: Kinetic Energy Distribution of Fragments

When a body at rest undergoes an explosion, the principles of conservation of momentum and the relationship between kinetic energy and mass play a crucial role in determining the kinetic energy of the resulting fragments. This article explores these principles, providing a detailed explanation and mathematical derivation to understand the distribution of kinetic energy in this scenario.

Conservation of Momentum and Kinetic Energy

The principle of conservation of momentum states that the total momentum of a closed system remains constant. In the case of an explosion, the system is initially at rest, meaning the total momentum is zero. As the body explodes into two fragments, the total momentum must still be zero. This condition allows us to establish a relationship between the velocities of the fragments.

Mathematical Derivation

In the case of two fragments, let the masses of the fragments be (m_1) and (m_2) such that (m_1 m) and (m_2 2m). Since the body was initially at rest, the initial momentum is zero. Therefore, the sum of the momenta of the fragments after the explosion must also be zero:

[begin{aligned} m_1v_1 - m_2v_2 0 m_1v_1 m_2v_2 v_1 frac{m_2}{m_1}v_2 text{Given } m_1 m text{ and } m_2 2m, v_1 2v_2end{aligned}]

Now, let's calculate the kinetic energies of the fragments. The kinetic energy of an object is given by:

[text{Kinetic Energy} frac{1}{2}mv^2]

For the first fragment:

[text{KE}_1 frac{1}{2}m_1v_1^2 frac{1}{2}m(2v_2)^2 frac{1}{2}m cdot 4v_2^2 2mv_2^2]

For the second fragment:

[text{KE}_2 frac{1}{2}m_2v_2^2 frac{1}{2}(2m)v_2^2 mv_2^2]

Finally, we can find the ratio of their kinetic energies:

[frac{text{KE}_1}{text{KE}_2} frac{2mv_2^2}{mv_2^2} 2]

Hence, the ratio of their kinetic energies is:

[boxed{2:1}]

Further Examples

Let's consider another scenario where the masses of the two bodies are assumed to be (m) and (2m). Substituting these values into the kinetic energy formula, we can verify that the ratio of kinetic energies remains the same:

[text{KE}_1 frac{1}{2}mv^2 quad text{and} quad text{KE}_2 frac{1}{2}(2m)v^2 mv^2]

[frac{text{KE}_1}{text{KE}_2} frac{frac{1}{2}mv^2}{mv^2} frac{1}{2}]

Hence, the ratio of the masses results in the inverse ratio of their kinetic energies, demonstrating the principle.

Conclusion

The scenario of an exploding body at rest illustrates the principles of conservation of momentum and kinetic energy. Understanding these principles is crucial for various applications, such as in ballistics, nuclear physics, and engineering mechanics. The precise calculations provided in this article can be applied to similar scenarios to predict and analyze the behavior of bodies in explosive situations.

Keywords

- kinetic energy

- explosion

- conservation of momentum

- body at rest

- fragment distribution

- mass ratio