## Tuesday, November 29, 2016

### Collisions Problems and Solutions Two

One dimensional elastic collision means bodies before and after collision travel in the same direction.  If both momentum and kinetic energy are conserved, the collision is said to be elastic collision. There is no wastage of energy in this case and all kinetic energy is conserved. In the case of one dimensional elastic collision, the velocity of approach of two bodies before collision is equal to the relative velocity of separation after the collision.This ratio is called coefficient of restitution and its value for elastic collision its value is one. In the case of perfect inelastic collision, the two bodies move together as one body and they have same velocity.

Collisions is a phenomenon where energy and momentum between two bodies are in interaction. There need not be physical contact for the collision. Change of path and transfer of linear momentum and kinetic energy is sufficient to say that the collision is happened. In the case of elastic collision, both kinetic energy and linear momentum are conserved. In the case of inelastic collision, only linear momentum is conserved but not kinetic energy. Some part of kinetic energy is converted into other formats of energy in this case. In the case of perfect inelastic collision, both the bodies after the collision move together and will have a common velocity.

Problem

A particle of mass one kilogram is thrown vertically upward with a speed of 100 meter per second.After five seconds it explodes into two parts of different masses. If velocity of one particle is known, we need to measure the velocity of the other part. Problem is as shown in the diagram below.

Solution

The body is moving against the acceleration due to gravity and we can find its velocity after five seconds using first equation of motion. We can also apply law of conservation of linear momentum and find the velocity of the second particle by applying proper sign convention. The solution is as shown in the diagram below.

Problem

It is given in the problem that a projectile of mass m is projected at an angle to the horizontal. At the maximum height of the projectile it breaks into two fragments of equal masses. One of the fragment retraces its path and we need to measure the velocity of the other piece.

Solution

At the maximum height of the projectile it has only horizontal component of the velocity and its vertical velocity component at that instant is zero. We can measure the initial momentum of the particle at the maximum height. The retraced particle has the same velocity but in the opposite direction. We can apply law of conservation of linear momentum and solve the problem as shown in the diagram below.

Problem

A particle of mass is moving with a velocity and it collides head on with another stationary particle and the collision is elastic. If the velocity of the second particle is known, we need to find the ratio of the mass of the two particles. Problem is as shown in the diagram below.

Solution

We can apply first law of conservation of linear momentum and find the relation between first and second velocities. We can also apply the concept of velocity of approach is the velocity of separation and get on more result. The solution is as shown in the diagram below.

Problem

It is given in the problem that a bullet of mass m moving with a velocity strikes a suspended wooden block of mass M. If the block rises to a height h, we need to measure the initial velocity of the bullet. Problem is as shown in the diagram below.

Solution

We can apply law of conservation of linear momentum to the given scenario and find the common velocity of the system when bullet is embedded into the block. This kind of collision is called perfect inelastic collision. We can equate the kinetic energy of the system into potential energy of the system and it can be further simplified as shown in the diagram below.

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