Prooving Elastic Collisions equations

In summary, an elastic collision is a collision in which the total kinetic energy of the colliding objects is conserved, resulting in no energy loss or deformation. The velocity of an object after an elastic collision can be calculated using a specific equation. This differs from an inelastic collision, where some kinetic energy is lost. Real-life examples of elastic collisions include billiard balls, basketballs, and molecules in a gas. The outcome of an elastic collision is impacted by factors such as the objects' masses, velocities, and external forces.
  • #1
Lord Dark
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Homework Statement


hi everyone ,, how are you all ? ,, i have these two equations and i don't know how to get them :
v1f=((m1-m2)/(m1+m2))*v1i +(2m2/(m1+m2))*v2i
V2f=((2m1/m1+m2))*v1i+((m2-m1)/(m1+m2))*v2i


Homework Equations


m1v1i+m2v2i=m1v1f+m2v2f (conservation of momentum)
0.5m1v1^2i+0.5m2v2i^2=0.5m1v1f^2+0.5m2v2f^2 (conservation of Kinetic Energy)


The Attempt at a Solution


the book reach to taking m1 & m2 as common factor then says divide the kinetic by the momentum then I'll get the results above, but when i divide i get :
v2f=v1i+v1f ,, so someone help me to get the results above because i don't like memorizing and i know that I'll forget in the exam -_-
 
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  • #2
m1v1i+m2v2i=m1v1f+m2v2f (conservation of momentum)
0.5m1v1^2i+0.5m2v2i^2=0.5m1v1f^2+0.5m2v2f^2 (conservation of Kinetic Energy)

In both the equations collect the terms containing m1 on one side and m2 on other side. Then divide left hand side and right hand side and equate. You will get relation between v1i, v1f ,v2i and v2f. Using this you can find the required result.
 
  • #3


I would say that the equations you have provided are the equations for elastic collisions, which follow the principles of conservation of momentum and conservation of kinetic energy. These equations are derived from the application of these principles to the system of colliding masses.

To prove these equations, we can start by considering the conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision. This can be expressed mathematically as:

m1v1i + m2v2i = m1v1f + m2v2f

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

Next, we consider the conservation of kinetic energy, which states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. This can be expressed mathematically as:

0.5m1v1i^2 + 0.5m2v2i^2 = 0.5m1v1f^2 + 0.5m2v2f^2

To prove the equations you have provided, we can manipulate these two equations. First, we can rearrange the conservation of momentum equation to solve for v1f and v2f:

v1f = (m1v1i + m2v2i - m2v2f) / m1
v2f = (m1v1i + m2v2i - m1v1f) / m2

Next, we can substitute these expressions for v1f and v2f into the conservation of kinetic energy equation:

0.5m1v1i^2 + 0.5m2v2i^2 = 0.5m1((m1v1i + m2v2i - m2v2f) / m1)^2 + 0.5m2((m1v1i + m2v2i - m1v1f) / m2)^2

Simplifying this equation, we get:

0.5m1v1i^2 + 0.5m2v2i^2 = 0.5m1v1i^2 + 0.5m2v2i^2
 

1. What is an elastic collision?

An elastic collision is a type of collision in which the total kinetic energy of the colliding objects is conserved. This means that no energy is lost during the collision, and the objects bounce off each other without any deformation.

2. How is the velocity of an object calculated in an elastic collision?

The velocity of an object after an elastic collision can be calculated using the equation: v = (m1u1 + m2u2) / (m1 + m2), where m1 and m2 are the masses of the objects and u1 and u2 are their initial velocities.

3. What is the difference between an elastic collision and an inelastic collision?

In an elastic collision, the total kinetic energy is conserved, whereas in an inelastic collision, some kinetic energy is lost in the form of heat, sound, or deformation of the objects involved.

4. Are there any real-life examples of elastic collisions?

Yes, there are many real-life examples of elastic collisions, such as the collision of billiard balls, bouncing of a basketball on the ground, or the collision of molecules in a gas.

5. What factors affect the outcome of an elastic collision?

The outcome of an elastic collision is affected by factors such as the masses and velocities of the objects involved, the angle and direction of their motion, and any external forces acting on them.

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