# 2D inelastic collision and K of CM

• sci-doo
In summary, we have two cars, one weighing 1500kg and traveling north at 80km/h and the other weighing 3500kg and traveling east at 50km/h. When they collide inelastically, the combined velocity of the wreck is 12m/s at an angle of 34.438 degrees. The kinetic energy lost in the collision is approximately 360 kJ. The kinetic energy of the center of mass for the two-car system remains the same before and after the collision, as stated by the principle of conservation of momentum. The lost kinetic energy is most likely converted into other forms, such as deformation and heat. The maximum energy loss for items sticking together after an inelastic collision is explained further on the
sci-doo

## Homework Statement

Car 1 (1500kg) is heading north 80km/h, Car 2 (3500kg) is heading east 50km/h. They collide inelastically.
a.) Speed and direction of the wreck (car 1 + car 2)?
b.) The kinetic energy lost?
HERE I HAVE A PROBLEM:
c.) What is the kinetic energy of the two-particle systems center of mass before and after the collision?
d.) Where was the lost kinetic energy hidden before the collision?

2. The attempt at a solution
a. is just vector sum p1+p2 where I can get the velocity by dividing it with the masses (about 12m/s), I can get the angle too (34,438... deg)
b. wrecks K minus K of car 1 and car 2 = - 360532,40... J so about 360 kJ lost
c. But this one I don't understand. I'd say it stays the same, because... well, that's what I calculated (CM from the v-vectors = v of CM), yet I don't really believe it...
d. ... after all shouldn't the K be lost to deformation of bodies (+heat etc.)? Does the K of CM stay the same anyways?

Ok, I found something useful on the net:
http://courses.washington.edu/ph122mo/A08/MT3info.html"
# Inelastic collisions

* momentum is still conserved
* kinetic energy of center of mass motion cannot be lost
* maximum energy loss for items sticking together after collision

And that is really all I needed to know.

Last edited by a moderator:

I would like to address the following points:

a. Your approach to calculating the speed and direction of the wreck is correct. However, it is important to note that the final velocity of the two cars will be a combination of both the initial velocities, rather than just the vector sum. This is because the collision is inelastic, meaning that some of the kinetic energy will be lost during the collision.

b. Your calculation of the kinetic energy lost is also correct. The lost kinetic energy is due to the deformation and damage to the cars during the collision. This energy is converted into other forms, such as heat and sound.

c. The kinetic energy of the two-particle system's center of mass (CM) before and after the collision will remain the same. This is because the CM is a property of the system as a whole, and it is not affected by internal interactions between the two cars. Therefore, even though some kinetic energy is lost during the collision, the total kinetic energy of the system (including the CM) will remain constant.

d. The lost kinetic energy is hidden in the deformation and damage to the cars, as well as in the form of heat and sound. As mentioned before, this energy is converted into other forms during the collision. The kinetic energy of the CM is not affected by this as it is a property of the system as a whole.

In summary, the conservation of energy is still applicable in an inelastic collision, but some of the kinetic energy is converted into other forms. The kinetic energy of the CM remains constant as it is a property of the system as a whole.

## 1. What is a 2D inelastic collision?

A 2D inelastic collision is a type of collision between two objects in which both momentum and kinetic energy are not conserved. In this type of collision, the two objects stick together after impact and move with a common velocity.

## 2. How is the center of mass (CM) calculated in a 2D inelastic collision?

The center of mass (CM) in a 2D inelastic collision can be calculated by finding the weighted average position of the two objects involved. This can be done by multiplying the mass of each object by its position and then dividing the sum by the total mass of the system.

## 3. What is the equation for kinetic energy (K) of the center of mass in a 2D inelastic collision?

The equation for the kinetic energy (K) of the center of mass in a 2D inelastic collision is K = ½ mvCM2, where m is the total mass of the system and vCM is the velocity of the center of mass.

## 4. How does the coefficient of restitution affect the kinetic energy of the center of mass in a 2D inelastic collision?

The coefficient of restitution (e) is a measure of the elasticity of a collision. In a 2D inelastic collision, where kinetic energy is not conserved, the value of e will affect the amount of kinetic energy lost during the collision. A higher value of e will result in less kinetic energy lost and a lower value of e will result in more kinetic energy lost.

## 5. Can the kinetic energy of the center of mass be negative in a 2D inelastic collision?

Yes, the kinetic energy of the center of mass can be negative in a 2D inelastic collision. This can occur when the two objects involved in the collision have opposite velocities and the resulting velocity of the center of mass is zero. In this case, the kinetic energy of the center of mass would be equal to -½ mvCM2.

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