## Collision - Momentum as mass times velocity - Use conservation of momentum principle to solve collision problems (elastic and inelastic collision) - Be able to calculate the kinetic energy in elastic and inelastic systems ### Momentum revision - Momentum is a measure of the quantity of motion possessed by a body - Momentum is a vector because velocity is a vector, so momentum will have magnitude and direction - Normally momentum is given as p - Units: kgm/s or Ns $ \text{Momentum}=\text{Mass}\times \text{Velocity} $ $ p=mv $ - Conservation of linear momentum: for any isolated system, the total linear momentum of the system is conserved. If you calculate the momentum before and after a collision, they should be the same. - Conservation of energy: for any isolated system, the total energy of the system is conserved. Energy before some event must be the same as after the event. ## Elastic and inelastic collisions Let's consider a one dimensional collision between two bodies. Ciollision may be elastic or inelastic. For completly inelastic colliusions, only linear momentum is conserved. Energy is lost in the form of heat, sound etc. $ \text{Completly Inelastic} (C_{R}=0)-\text{KE loss, object stick} $ In a completly elastic collision, bot the linear momentum and the kinetic energy of the system as conserved. $ \text{Completly Inelastic} (C_{R}=1)-\text{KE conserved} $ Elasticity or coefficient of restitution: $ C_{R}=\left|\frac{v_{1}-v_{2}}{u_{1}-u_{2}}\right| $ Consider two bodies have masses $m_{1}$ and $m_{2}$. We are dealing with completely inelastic collision, so we imagine that these masses are covered with glue, such that when colliding they stick together. ## Inelastic collision $ p_{before}=p_{after} $ Just before the impact, the masses had total momentum (scalar version): $ p_{before}=m_{1}u_{1}+m_{2}u_{2} $ Immediately after the impact, the momentum has changed to: $ p_{after}=(m_{1}+m_{2})v $ Uf we know the initial condition and the masses, then v can be found as: $ v=\frac{m_{1}u_{1}+m_{2}u_{2}}{(m_{1}+m_{2})} $ If we put all the non kinetic energies together, we can reqrite the energy balance for before and after the collision: $ KE_{1}=\text{Energy Loss}+KE_{2} $ In inelastic collision, Energy Loss < 0, kinetic energy has been lost from the system (normally as heat or sound) and results in an inelastic collision. TODO: KE equations ## Elastic collision For elastic collision, momentum before and after are the same. Just before the impact, the masses had a total momentum: $ p_{before}=m_{1}u_{1}+m_{2}u_{2} $ After the impact, the momentum is $ p_{after}=m_{1}v_{1}+m_{2}v_{2} $ In elastic collision, energy loss is zero. Kinetic energy is conserved, there is no energy loss.