Calculating the Final Velocity of Mass 1

Oct 13, 2025 by wbrum.org 100 views

Introduction

In physics, collisions between objects are a fundamental concept that helps us understand the behavior of particles and objects in various situations. An elastic collision is a type of collision where the total kinetic energy of the system is conserved, and the objects involved return to their original shape after the collision. In this article, we will explore an elastic collision between two masses and calculate the final velocity of mass 1.

Given Information

We are given two masses:

Mass 1 (m1) = 0.311 kg
Mass 2 (m2) = 0.0570 kg

The initial velocities of the masses are:

v1i = +30.3 m/s (mass 1 is moving in the positive direction)
v2i = -19.2 m/s (mass 2 is moving in the negative direction)

Conservation of Momentum

In an elastic collision, the total momentum of the system is conserved. We can use the following equation to calculate the final velocity of mass 1:

m1v1i + m2v2i = m1v1f + m2v2f

where v1f is the final velocity of mass 1, and v2f is the final velocity of mass 2.

Calculating the Final Velocity of Mass 1

We can rearrange the equation to solve for v1f:

v1f = (m1v1i + m2v2i - m2v2f) / m1

To find v2f, we can use the fact that the total kinetic energy of the system is conserved. We can write an equation for the initial kinetic energy and the final kinetic energy:

(1/2)m1v1i^2 + (1/2)m2v2i^2 = (1/2)m1v1f^2 + (1/2)m2v2f^2

We can simplify this equation by substituting the values of m1, m2, v1i, and v2i:

(1/2)(0.311 kg)(30.3 m/s)^2 + (1/2)(0.0570 kg)(-19.2 m/s)^2 = (1/2)(0.311 kg)v1f^2 + (1/2)(0.0570 kg)v2f^2

We can solve this equation for v2f:

v2f = (-0.0570 kg)(-19.2 m/s)^2 / (0.0570 kg) + (0.311 kg)(30.3 m/s)^2 / (0.311 kg) - (0.0570 kg)(-19.2 m/s)^2 / (0.0570 kg)

v2f = 0.0570 kg(368.64 m^2/s2) / 0.0570 kg + 0.311 kg(918.09 m^2/s2) / 0.311 kg - 0.0570 kg(368.64 m^2/s2) / 0.0570 kg

v2f = 20.8 m/s + 2951.1 m/s - 20.8 m/s

v2f = 2931.1 m/s

Now that we have v2f, we can substitute it into the equation for v1f:

v1f = (m1v1i + m2v2i - m2v2f) / m1

v1f = (0.311 kg)(30.3 m/s) + (0.0570 kg)(-19.2 m/s) - (0.0570 kg)(2931.1 m/s) / 0.311 kg

v1f = 9.45 m/s + (-1.0956 m/s) - 938.3 m/s

v1f = -930.1 m/s

Conclusion

In this article, we calculated the final velocity of mass 1 after an elastic collision between two masses. We used the conservation of momentum and the conservation of kinetic energy to solve for the final velocity of mass 1. The final velocity of mass 1 is -930.1 m/s.

Discussion

The final velocity of mass 1 is negative, indicating that it is moving in the negative direction after the collision. This is because mass 2 has a larger mass and a larger initial velocity, causing mass 1 to be deflected in the opposite direction.

Limitations

This calculation assumes that the collision is perfectly elastic, meaning that the total kinetic energy of the system is conserved. In reality, collisions are often inelastic, meaning that some of the kinetic energy is converted into other forms of energy, such as heat or sound.

Future Work

In future work, we can explore more complex collisions, such as inelastic collisions or collisions involving multiple objects. We can also use numerical methods to solve for the final velocities of the masses.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.
Elastic Collision of Two Masses: Q&A =====================================

Q: What is an elastic collision?

A: An elastic collision is a type of collision where the total kinetic energy of the system is conserved, and the objects involved return to their original shape after the collision.

Q: What is the difference between an elastic collision and an inelastic collision?

A: In an inelastic collision, some of the kinetic energy is converted into other forms of energy, such as heat or sound. In an elastic collision, the total kinetic energy of the system is conserved.

Q: How do you calculate the final velocity of mass 1 after an elastic collision?

A: To calculate the final velocity of mass 1, you can use the conservation of momentum equation:

m1v1i + m2v2i = m1v1f + m2v2f

where v1f is the final velocity of mass 1, and v2f is the final velocity of mass 2.

Q: What is the formula for the final velocity of mass 1?

A: The formula for the final velocity of mass 1 is:

v1f = (m1v1i + m2v2i - m2v2f) / m1

Q: How do you calculate the final velocity of mass 2 after an elastic collision?

A: To calculate the final velocity of mass 2, you can use the conservation of kinetic energy equation:

(1/2)m1v1i^2 + (1/2)m2v2i^2 = (1/2)m1v1f^2 + (1/2)m2v2f^2

Q: What is the formula for the final velocity of mass 2?

A: The formula for the final velocity of mass 2 is:

v2f = (-0.0570 kg)(-19.2 m/s)^2 / (0.0570 kg) + (0.311 kg)(30.3 m/s)^2 / (0.311 kg) - (0.0570 kg)(-19.2 m/s)^2 / (0.0570 kg)

Q: What is the final velocity of mass 1 after an elastic collision between two masses?

A: The final velocity of mass 1 is -930.1 m/s.

Q: Why is the final velocity of mass 1 negative?

A: The final velocity of mass 1 is negative because mass 2 has a larger mass and a larger initial velocity, causing mass 1 to be deflected in the opposite direction.

Q: What are some limitations of this calculation?

A: This calculation assumes that the collision is perfectly elastic, meaning that the total kinetic energy of the system is conserved. In reality, collisions are often inelastic, meaning that some of the kinetic energy is converted into other forms of energy, such as heat or sound.

Q: What are some future directions for this research?

A: Some future directions for this research include exploring more complex collisions, such as inelastic collisions or collisions involving multiple objects. We can also use numerical methods to solve for the final velocities of the masses.

Q: What are some real-world applications of this research?

A: This research has real-world applications in fields such as physics, engineering, and materials science. Understanding the behavior of particles and objects in collisions is crucial for designing and optimizing systems, such as crash test dummies, car airbags, and impact-resistant materials.