## One-dimensional

## Elastic and Inelastic Collisions

Of course, collisions come in many forms. We will restrict our attention to two very special kinds.

Inelastic collisionsare those in which the two objects stick together.Consider a situation with mass m

_{1}coming in with initial velocityv_{1i}and striking mass m_{2}that isat rest(v_{2i}= 0).The two masses

stick togetherso this is atotally inelastic collision.The

initialmomentum is given byP _{Tot,i }= p_{1i}+ p_{2i}P

_{Tot,i}= m_{1}v_{1i}

Afterthe collision, thefinalmomentum is given byP _{Tot,f}= (m_{1}+ m_{2}) v_{f}Since momentum is conserved, this gives

P _{Tot,f}= P_{Tot,i}(m

_{1}+ m_{2}) v_{f}= m_{1}v_{1i}v

_{f}= [ m_{1}/ (m_{1}+ m_{2}) ] v_{1i}

Elasticcollisionsare those in which the Kinetic Energy is also conserved.Consider two objects that "interact" or collide or have some effect on each other -- in such a way that the total Kinetic Energy is also conserved. This is a totally

elasticcollision.P

_{Tot,i}= P_{Tot,f}m

_{1}v_{1i }+ m_{2}v_{2i}= m_{1}v_{1f}+ m_{2}v_{2f}Remember,

momentum is a vector!That means that a velocity to the right ispositiveand a velocity to the left isnegative.Since this is an

elasticcollision, wealsohaveK _{Tot,i}= K_{Tot,f}(

^{1}/_{2}) m_{1}v_{1i}^{2}+ (^{1}/_{2}) m_{2}v_{2i }^{2}= (^{1}/_{2}) m_{1}v_{1f}^{2}+ (^{1}/_{2}) m_{2}v_{2f }^{2}This provides

two equationsfrom which we can solve for thetwo unknowns, v_{1f }and v_{2f }.

Aspecial casethat is interesting to look at is that of an "incoming" object of mass m_{1}with initial velocity v_{1i}that collideselasticallywith mass m_{2}that isat rest. That is v_{2i}= 0.The equations for the final velocities are

and

Notice what happens when the masses are equal; for m

_{1}= m_{2}:And notice for m

_{1}> m_{2}:And notice for m

_{1}< m_{2}:These

elasticcollisions are consistent with -- they are predicted by -- these equations:and

I have some

animationfor these collisions on my web site for PHY 3050 which I have not yet incorporated into this web site. Click here for thatanimation.PHY 3050 has a lighter description of

SPLAT - BOING!problems that you might enjoy.

Collisions2D CollisionsReturn to ToC, Ch9, Linear Momentum(c) Doug Davis, 2001; all rights reserved