(II) A 6.0 -kg object moving in the +x direction at 5.5 m / s collides head-on with an 8.0-kg ob

(II) A 6.0 -kg object moving in the +x direction at 5.5 m /...

(II) A 6.0 -kg object moving in the $+x$ direction at 5.5 $\mathrm{m} / \mathrm{s}$ collides head-on with an $8.0-\mathrm{kg}$ object moving in the $-x$ direction at 4.0 $\mathrm{m} / \mathrm{s} .$ Find the final velocity of each mass if: $(a)$ the objects stick together; $(b)$ the collision is elastic; $(c)$ the $6.0-\mathrm{kg}$ object is at rest after the collision; $(d)$ the 8.0 -kg object is at rest after the collision; $(e)$ the 6.0 -kg object has a velocity of 4.0 $\mathrm{m} / \mathrm{s}$ in the $-x$ direction after the collision. Are the results in $(c),(d),$ and $(e)$ "reasonable"? Explain.

Key Concepts

Analysis of Special Velocity Constraints

When certain conditions are imposed on collisions—for example specifying that one object comes to rest or moves with a given velocity—one must check these outcomes against the conservation laws. Imposing such constraints can sometimes lead to results that violate the simultaneous conservation of momentum and energy, indicating that they may be physically unreasonable or represent external forces acting on the system.

Conservation of Linear Momentum

This principle states that the total momentum of a closed system remains constant if no external forces act on it. In collision problems, regardless of whether the collision is elastic or inelastic, the sum of the momenta before the collision is equal to the sum of the momenta after the collision. This concept is central to solving for unknown velocities when objects interact through collisions.

Completely Inelastic Collisions

A completely inelastic collision is one in which the colliding objects stick together after impact. While momentum is conserved in these collisions, kinetic energy is not necessarily conserved. This concept simplifies the analysis since the final state is characterized by a single velocity for both masses combined.

Elastic Collisions

Elastic collisions are those in which both momentum and kinetic energy are conserved throughout the interaction. In these collisions, the objects rebound off each other without losing energy to deformation or heat. This dual conservation results in a set of equations that can be solved to find the post-collision velocities.

Conservation of Kinetic Energy

The conservation of kinetic energy applies to elastic collisions, where not only momentum but also kinetic energy remains constant before and after the collision. This principle is used alongside momentum conservation to determine the final velocities of the objects in scenarios where no energy is dissipated.

Transcript

00:01 Two balls a and b, a having mass 6 kilogram and b having mass 8 kilogram, moving with velocities 5 .5 meter per second in the positive x direction, a and b is moving in the negative x direction with 4 meter per second velocity.

00:19 The question is, after colliding, they stick together.

00:25 Then what is the velocity after the collision? so that is our first question.

00:32 So if the ball stick together, it is a completely inelastic collision.

00:41 So we cannot apply law of conservation of energy, but momentum is still conserved.

00:48 So we can apply conservation of momentum.

00:53 That is, we can write momentum before collision is equal to momentum after collision.

01:03 This m a into va plus mb.

01:08 Into v b is equal to final the mass both the masses stick together so the final mass will be m a plus m b into v we have to find this v so v equal to rearranging this m a plus m b into v b by m a plus m a plus m a plus m b which is equal to six into 5 .5 and this 4 will be treated as minus 4 meter per second because it is the opposite direction since velocity is a vector minus 8 into the 4.

01:57 The minus 9 belongs to the 4 divided by total masses 6 plus 8 which is 14 which is equal to 7 .1 into 10 raise 2 minus 2 meter per second.

02:13 Now the second case which is completely elastic collation.

02:22 So you have to find the final velocities of the balls.

02:29 So we can derive the expressions bb prime using law of conservation of energy and law of conservation of momentum.

02:40 Both the conservation laws state that the total energy or total momentum before coalition is equal to total momentum or total momentum or total energy...

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