10 Elastic Collision Formulas For Easy Calculations

Elastic collisions are a fundamental concept in physics, where two objects collide and rebound without losing any kinetic energy. The study of elastic collisions involves the use of various formulas to calculate the velocities of the objects before and after the collision. In this article, we will explore 10 essential elastic collision formulas that can simplify calculations and provide a deeper understanding of the underlying physics.

Introduction to Elastic Collisions

Elastic collisions occur when two objects collide and separate without any loss of kinetic energy. This means that the total kinetic energy of the system remains constant before and after the collision. The formulas used to describe elastic collisions are based on the principles of conservation of momentum and conservation of kinetic energy.

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant over time. In an elastic collision, the momentum of the objects before the collision is equal to the momentum of the objects after the collision. This can be expressed using the following formula:

m1v1i + m2v2i = m1v1f + m2v2f

where m1 and m2 are the masses of the objects, v1i and v2i are the initial velocities, and v1f and v2f are the final velocities.

10 Elastic Collision Formulas

The following formulas are used to calculate the velocities of the objects involved in an elastic collision:

1. Velocity of the first object after collision: v1f = (v1i(m1 - m2) + 2m2v2i) / (m1 + m2)
2. Velocity of the second object after collision: v2f = (v2i(m2 - m1) + 2m1v1i) / (m1 + m2)
3. Relative velocity before collision: vrel,i = v1i - v2i
4. Relative velocity after collision: vrel,f = v1f - v2f
5. Conservation of kinetic energy: (1/2)m1v1i^2 + (1/2)m2v2i^2 = (1/2)m1v1f^2 + (1/2)m2v2f^2
6. Conservation of momentum in one dimension: m1v1i + m2v2i = m1v1f + m2v2f
7. Conservation of momentum in two dimensions: m1v1i,x + m2v2i,x = m1v1f,x + m2v2f,x and m1v1i,y + m2v2i,y = m1v1f,y + m2v2f,y
8. Collision in one dimension with equal masses: v1f = v2i and v2f = v1i
9. Collision in one dimension with unequal masses: v1f = (v1i(m1 - m2) + 2m2v2i) / (m1 + m2) and v2f = (v2i(m2 - m1) + 2m1v1i) / (m1 + m2)
10. Collision in two dimensions with equal masses: v1f,x = v2i,x and v1f,y = v2i,y and v2f,x = v1i,x and v2f,y = v1i,y

Example Calculation

Suppose we have two objects, A and B, with masses 2 kg and 3 kg, respectively. Object A is moving at a velocity of 4 m/s, while object B is moving at a velocity of 2 m/s. We can use the formulas above to calculate the velocities of the objects after the collision.

First, we calculate the velocity of object A after the collision using formula 1:

v1f = (v1i(m1 - m2) + 2m2v2i) / (m1 + m2) = (4(2 - 3) + 2*3*2) / (2 + 3) = (-4 + 12) / 5 = 8/5 = 1.6 m/s

Next, we calculate the velocity of object B after the collision using formula 2:

v2f = (v2i(m2 - m1) + 2m1v1i) / (m1 + m2) = (2(3 - 2) + 2*2*4) / (2 + 3) = (2 + 16) / 5 = 18/5 = 3.6 m/s

These calculations demonstrate how the formulas can be used to determine the velocities of the objects after an elastic collision.

Conclusion

In conclusion, the 10 elastic collision formulas provided in this article can be used to calculate the velocities of objects involved in an elastic collision. By understanding and applying these formulas, physicists and engineers can gain a deeper insight into the underlying physics of collisions and make accurate predictions about the behavior of objects in various scenarios.