A perfectly inelastic collision is one in which two objects colliding stick together, becoming a single object. For instance, two balls of sticky putty thrown at each other would likely result in perfectly inelastic collision: the two balls stick together and become a single object after the collision.
Unlike elastic collisions, perfectly inelastic collisions don't conserve energy, but they do conserve momentum. While the total energy of a system is always conserved, the kinetic energy carried by the moving objects is not always conserved. In an inelastic collision, energy is lost to the environment, transferred into other forms such as heat.
and with the same mass are m away from each other. Now, and are thrown horizontally at the same time at the velocities m/s and m/s, respectively, eventually colliding with each other in the air. If the two objects stick together after a perfectly inelastic collision, what is the speed of the mass at the moment of collision (in m/s)?In the above figure, two objects
Assume that gravitational acceleration is m/s.
Consider two particles of mass and moving at velocities and , respectively. Before they collide, they have a combined energy of and a combined momentum of .
Since the collision is perfectly inelastic, after the collision there is a single combined object of mass . Since momentum is conserved, this object has momentum equal to the total intitial momentum . The velocity of the combined object is then given by
The energy depends on the squared magnitude of , which is the dot product of with itself. If the angle between and is , then this equals
The final energy is
This equation is the general solution for perfectly inelastic collisions. It's somewhat ugly, but exploring how it works in particular simplified cases can help build intuition for what it says.
What is the energy difference if is much much smaller than ? Physicists express this with symbols as . In this case, . This simplifies the equation to
Since , , the last term is small if in addition is smaller than or not much larger than . These combined assumptions allow to be further simplified to
This equation gives a nice interpretation for this limiting case. The second term "eliminates" the energy of the original particle, while the first term "creates" a particle of mass with velocity projected in the direction of the more massive , because it's stuck to . The energy of the mass is left unchanged.
Take special care that this simplification required that the velocity of the smaller particle was not too high. If it were, then the larger particle would have its energy changed as well, and