Lecture 16: Momentum, Collisions

Overview

PHY231, Introductory Physics I, focuses on momentum and collisions. This lecture specifically highlights the concepts of elastic and inelastic collisions.

Types of Collisions

Elastic collisions are defined as collisions where both total momentum and total kinetic energy are conserved. Mathematically, this is represented by $p{\text{total,f}} = p{\text{total,i}}$ and $KE{\text{total,f}} = KE{\text{total,i}}$ . In contrast, inelastic collisions are defined as collisions where total momentum is conserved, but total kinetic energy is not conserved, as some kinetic energy is transformed into thermal energy. For perfectly inelastic collisions, the objects stick together after the collision. The mathematical representation for momentum conservation in these cases is $p{\text{total,f}} = p{\text{total,i}}$ . It is important to note that most real-world collisions fall into intermediate cases, existing somewhere between perfectly elastic and perfectly inelastic types.

Inelastic Collisions (1D)

In the context of a one-dimensional inelastic collision between two objects, only the conservation of momentum applies. For perfectly inelastic collisions, where objects stick together, their final velocity is the same. The conservation of momentum can be expressed as $p{\text{total,f}} = p{\text{total,i}}$ . Specifically, the momentum before collision is given by $m1 v{1i} + m2 v{2i}$ , and the momentum after collision is $(m1 + m2) vf$ . Combining these expressions yields the equation $(m1 + m2)vf = m1 v{1i} + m2 v{2i}$ , which can be rewritten to solve for the final velocity as $vf = \frac{m1 v{1i} + m2 v{2i}}{m1 + m_2}$ .

Elastic Collisions (1D)

For one-dimensional elastic collisions, both the conservation of momentum and the conservation of kinetic energy apply. The momentum equation is $p{\text{total,f}} = p{\text{total,i}}$ , which translates to $m1 v{1f} + m2 v{2f} = m1 v{1i} + m2 v{2i}$ . Concurrently, the kinetic energy equation is $KE{\text{total,f}} = KE{\text{total,i}}$ , expressed as $\frac{1}{2} m1 v{1f}^2 + \frac{1}{2} m2 v{2f}^2 = \frac{1}{2} m1 v{1i}^2 + \frac{1}{2} m2 v{2i}^2$ .

Summary of Conservation Principles

Momentum is conserved for both elastic and inelastic collision types, represented by $p{\text{total,f}} = p{\text{total,i}}$ . However, kinetic energy conservation differs: it is conserved in elastic collisions ( $KEi = KEf$ ) but not conserved in inelastic collisions.

Elastic Collision Demonstration (1)

Consider a scenario involving two carts, each with mass $m$ , on a frictionless rail system, moving in opposite directions with the same speed. The initial velocities are antagonistic, i.e., $v{2i} = -v{1i}$ . For such an elastic collision, the conservation of momentum is expressed as $m1 v{1f} + m2 v{2f} = m1 v{1i} + m2 v{2i}$ .

Results from Demonstration (1)

In the simplified case where identical masses move with equal speeds in opposite directions, analyzing both momentum and energy conservation yields the solution: $v{1f} = -v{2f}$ , and subsequently, $v{1f} = vi, \text{ } v{2f} = vi$ .

Elastic Collision Demonstration (2)

Another demonstration involves a head-on elastic collision between two objects of different masses, specifically $m$ and $3m$ . Both objects possess the same initial speed. The objective is to determine the velocity of the $3m$ object after the collision. Momentum conservation leads to the equation $mv1 + 3mv2 = mv{1i} + 3mv{2i}$ . Kinetic energy conservation further establishes relationships between the speeds before and after the collision. The final results indicate that $v{1f} = -2vi, \text{ and } v_{2f} = 0$ .

Center of Mass

The center of mass ( $r{\text{CM}}$ ) represents the weighted average position of all mass within an object. It is calculated for the x-direction ( $x{\text{CM}}$ ) and y-direction ( $y{\text{CM}}$ ) using the total position formula: $r{\text{CM}} = \frac{\text{sum of } (mj xj)}{\text{sum of } mj}$ . Similarly, for the y-direction, it is formulated as $y{\text{CM}} = \frac{\text{sum of } (mj yj)}{\text{sum of } m_j}$ .

Example: Center of Mass

If two unequal masses, $m1$ and $m2$ , are connected by a light rod, and mass $m2$ is significantly greater than $m1$ , the center of mass would be located closer to $m2$ . The given options were: 1. In the middle of the rod, 2. Closer to $m1$ , 3. Closer to $m_2$ , 4. Impossible to say.

Work-Energy Theorem

The Work-Energy Theorem provides an equation for non-conservative forces: $W_{nc} = \Delta KE + \Delta PE$ . This formula allows for the consideration of changes in both kinetic energy (KE) and potential energy (PE), stating that the total work done by non-conservative forces equals the change in kinetic energy plus the change in potential energy.