# Problem with inelastic collision

• ShayanJ
In summary, the conversation discussed the calculation of kinetic energy loss after a collision between two masses, m1 and m2, with velocities v1 and v2, respectively. The coefficient of restitution was given as e and the problem was solved in the center of mass coordinates. The relative speeds before and after the collision were found to be vr = v2 - v1 and ur = u2 - u1, respectively. However, the relative speeds were initially calculated as vr = -(v1 + v2) and ur = u1 + u2, leading to incorrect results. After correcting these mistakes, the correct formula for the kinetic energy loss, Q, was found to be Q = (m1m2/2)(m1
ShayanJ
Gold Member
Two masses $m_1$ and $m_2$ are closing each other with speeds $v_1$ and $v_2$. The coefficient of restitution is e. Calculate the amount of kinetic energy loss after caused by the collision.
I solved it in the center of mass coordinates($v_{cm}=u_{cm}=0$). The relative speed before and after the collision are $v_r=-(v_1+v_2)$ and $u_r=u_1+u_2$ respectively. Using conservation of momentum, we know that $m_1v_1=-m_2v_2$ and $m_1u_1=-m_2u_2$. Solving these equations for $v_1,v_2,u_1,u_2$, we'll have:
$v_1=-\frac{m_2}{m_2-m_1}v_r\\ v_2=\frac{m_1}{m_2-m_1}v_r\\ u_1=\frac{m_2}{m_2-m_1}u_r\\ u_2=-\frac{m_1}{m_2-m_1}u_r$
Substituting the above results into $m_1v_1^2+m_2v_2^2=m_1u_1^2+m_2u_2^2+2Q$ and using $u_r=e v_r$, We'll have:
$Q=\frac{m_1m_2}{2} \frac{m_1+m_2}{(m_1-m_2)^2} (1-e^2) v_r^2$
But as you can see, this is saying that for $m_1=m_2$ , Q becomes infinite which has no meaning and so something must be wrong. But I can't find what is that. What is it?
Thanks

Last edited:
You use two different conventions for the speeds/velocities - for the relative speed, you use them as absolute values to add them, but in the conservation of momentum, you use them as vectors (which can be negative).

It is easier to use them as velocity, then your relative speed is wrong.

Your relative speeds are wrong. They should be vr = v2 - v1, and ur = u2 - u1 respectively

EDIT: I see that mfb beat me to the punch

mfb said:
You use two different conventions for the speeds/velocities - for the relative speed, you use them as absolute values to add them, but in the conservation of momentum, you use them as vectors (which can be negative).

It is easier to use them as velocity, then your relative speed is wrong.
Ohh...yeah...thanks man.
Sometimes I really think I have some problems in the basics!

dauto said:
Your relative speeds are wrong. They should be vr = v2 - v1, and ur = u2 - u1 respectively
That's when you're dealing them as vectors. When you're dealing with their components, negative signs may appear which may alter that formula.

for your question. The problem with this calculation is that it assumes a perfectly inelastic collision, where the two masses stick together after the collision. In reality, there will always be some loss of kinetic energy due to factors such as friction and deformation of the objects involved. This is why the coefficient of restitution, e, is used to account for this loss of kinetic energy.

In your calculation, you have assumed that the coefficient of restitution is 1, which would mean a perfectly elastic collision. However, in an inelastic collision, the coefficient of restitution is always less than 1. This means that some of the kinetic energy is lost in the form of heat, sound, and deformation.

In order to accurately calculate the kinetic energy loss in an inelastic collision, you would need to know the specific properties of the objects involved, such as their elasticity, surface roughness, and any other factors that could affect the collision. Without this information, it is not possible to accurately calculate the amount of kinetic energy lost.

In summary, the problem with your calculation is that it does not take into account the loss of kinetic energy in an inelastic collision. To accurately calculate this, you would need to know the specific properties of the objects involved and use a more complex equation that takes into account the coefficient of restitution and other factors.

## What is an inelastic collision?

An inelastic collision is a type of collision in which the objects involved do not rebound or bounce off each other. Instead, they stick together and move as one object after the collision. This type of collision is characterized by a loss of kinetic energy.

## What is the difference between inelastic and elastic collisions?

In an elastic collision, the objects involved do rebound or bounce off each other after the collision. This type of collision is characterized by conservation of kinetic energy. In an inelastic collision, the objects stick together and move as one object, resulting in a loss of kinetic energy.

## What causes a problem with inelastic collision?

A problem with inelastic collision can occur when there is an external force acting on the objects involved, causing them to stick together. This can result in a loss of kinetic energy and can make it difficult to accurately calculate the final velocities of the objects.

## How can one solve a problem with inelastic collision?

To solve a problem with inelastic collision, it is important to first identify the external forces acting on the objects involved. Then, use the conservation of momentum and energy principles to calculate the final velocities of the objects. It may also be helpful to use equations specific to inelastic collisions, such as the coefficient of restitution.

## What are some real-life examples of inelastic collisions?

Inelastic collisions occur frequently in everyday life. Some examples include a car accident, where the cars stick together and move as one object after the collision, or a ball hitting the ground and not rebounding back to its original height. Inelastic collisions also occur in sports, such as when a baseball player catches a ball and their hand and the ball move together after the catch.

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