2.9 Thermodynamics and Inelastic Collisions: Conservation of Momentum

In the last section, we talked about what momentum was. Then we listed the two types of collisions: elastic and inelastic. We defined and discovered more about elastic collisions. In this section, we will dive in and take a closer look at inelastic collisions! 🧠 

Inelastic Collisions

If the initial kinetic energy equals the final kinetic energy, then the objects went through an elastic collision (rebound) since KE was conserved and so was momentum. In situations where KE isn’t conserved during a collision due to losses associated with heat, sound, etc, that type is known as an inelastic collision (stick together). In either type of collision, momentum will be conserved but the KE may not be so it’s important to be able to identify which collision is occurring in the problem. In completely inelastic collisions, the objects stick together after the collision after colliding. 

Example: Two carts of the same mass lie on a table. The first cart is moving and the second cart is at rest. At the end of the collision, the two carts were stuck together. What is the final speed of the system? To find the initial momentum of two carts, do pi = m1v1 + m2v2. To find the final momentum of two carts that stick together after the collision, do pf= (m1+ m2) vf. You can set the initial equation equal to the pfinal equation since momentum is conserved. Finally, solve for the final velocity. 

A lot of the collisions that happen on the molecular level are somewhat inelastic. We tend to ignore this and assume collisions are elastic because the we just want an approximate model.

But it is important to remember the difference between inelastic and elastic collisions! 

Key points about the difference between inelastic and elastic collisions:

• In a collision, two objects interact and exchange energy and momentum.
• In an elastic collision, the total kinetic energy of the colliding objects is conserved. This means that the kinetic energy before the collision is equal to the kinetic energy after the collision.
• In an inelastic collision, some of the kinetic energy is transformed into other forms of energy, such as thermal energy or sound energy. This means that the kinetic energy before the collision is not equal to the kinetic energy after the collision.
• In an elastic collision, the colliding objects will bounce off each other with the same speed and direction they had before the collision. In an inelastic collision, the colliding objects will stick together or deform and continue moving together as a single object after the collision.
• The type of collision that occurs depends on the properties of the colliding objects and the nature of the collision. For example, collisions between hard, non-deformable objects tend to be more elastic, while collisions between soft, deformable objects tend to be more inelastic.

Example Problem #1:

Two carts, one with a mass of 5 kg and the other with a mass of 2 kg, collide on a frictionless track. The 5 kg cart is initially moving at a speed of 3 m/s to the right, and the 2 kg cart is initially stationary. After the collision, the 5 kg cart moves to the left at a speed of 1 m/s.

(a) Is this collision elastic or inelastic? Justify your answer. (b) What is the common final velocity of the carts after the collision? (c) What is the initial kinetic energy of the 5 kg cart? (d) What is the final kinetic energy of the 5 kg cart? (e) What is the change in kinetic energy of the 5 kg cart during the collision?"

To solve this problem, you would first need to classify the collision as either elastic or inelastic. To do this, you could use the fact that in an elastic collision, the total kinetic energy of the colliding objects is conserved. In this case, the initial kinetic energy of the 5 kg cart is 3^25 = 45 J, and the final kinetic energy of the 5 kg cart is 1^25 = 5 J, which are not equal. Therefore, this collision is inelastic.

Next, you could use the conservation of linear momentum to find the common final velocity of the carts after the collision. To do this, you would set the initial momentum of the 5 kg cart to the final momentum of the 2 kg cart, and solve for the final velocity of the 2 kg cart. This would give you a common final velocity of -1 m/s for the carts.

Then, you could use the initial and final velocities and masses of the carts to solve for the initial and final kinetic energies of the 5 kg cart and the change in kinetic energy during the collision. This would give you an initial kinetic energy of 45 J, a final kinetic energy of 5 J, and a change in kinetic energy of -40 J during the collision.

Example Problem #2:

A 20 kg bowling ball traveling at a speed of 5 m/s to the left collides with a stationary 10 kg bowling ball on a frictionless alley. After the collision, the 20 kg ball moves at a speed of 2 m/s to the right, and the 10 kg ball moves at a speed of 3 m/s to the left.

(a) Is this collision elastic or inelastic? Justify your answer. (b) What is the change in kinetic energy of the system during the collision?

To solve this problem, you would first need to classify the collision as either elastic or inelastic. To do this, you could use the fact that in an elastic collision, the total kinetic energy of the colliding objects is conserved. In this case, the initial kinetic energy of the 20 kg ball is 5^220 = 500 J, and the final kinetic energy of the combined balls is (2^220)+(3^2*10) = 280 J, which are not equal. Therefore, this collision is inelastic.

Next, you could use the conservation of linear momentum to find the change in kinetic energy of the system during the collision. To do this, you would set the initial momentum of the 20 kg ball to the final momentum of the combined balls, and solve for the final kinetic energy of the combined balls. This would give you a change in kinetic energy of 220 J during the collision.