In studying energy during collisions, we look at two main types: elastic and inelastic collisions. Each type has different energy features that we can describe with math to help predict what happens when things collide.

Elastic Collisions

In elastic collisions, both kinetic energy and momentum are kept the same. Here’s how we can write this in simple math:

1. Momentum Conservation:

   $$m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$$

2. Kinetic Energy Conservation:

   $$\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$$

These equations help us find out how fast both objects will be moving after they hit each other.

Inelastic Collisions

In inelastic collisions, momentum is still conserved, but kinetic energy is not. This means that some of the kinetic energy changes into other types of energy, like heat or sound. Let’s look at the math:

1. Momentum Conservation:

   $$m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$$

2. Loss of Kinetic Energy: The change in kinetic energy can be shown, but it doesn’t stay the same. Usually, we calculate it like this:

   $$\Delta KE = KE_{initial} - KE_{final}$$

In a perfectly inelastic collision, the two objects stick together after colliding. The final speed, $v_f$, can be found using this formula:

$$v_f = \frac{m_1v_{1i} + m_2v_{2i}}{m_1 + m_2}$$

Conclusion

These math models help us understand how collisions work and how energy changes during these events. Knowing these ideas is important for tackling more challenging problems in University Physics I.