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  Linear Collision, Physics tutorial

Classification of Collisions:

There are two kinds of collisions namely elastic and inelastic collisions. Elastic collision is the collision between two or more objects in which no energy is lost. I.e., total kinetic energy of objects before collision is equal to total kinetic energy of objects after collision. Kinetic energy is conserved. But if kinetic energy is not conserved in the collision the collision is known as inelastic collision. This implies that in inelastic collision, some kinetic energy is converted to heat or sound. There is also the situation in which two bodies can collide and coalesce (that is stick together). This type of collision is referred to as perfectly inelastic collision as it corresponds to the situation where maximum kinetic energy is lost in collision.

Perfectly Inelastic Collision:

A perfectly elastic collision is stated as one in which there is no loss of kinetic energy in collision. The inelastic collision is one in which part of kinetic energy is changed to some other form of energy in collision. Any macroscopic collision between objects will convert some kinetic energy in internal energy and other forms of energy, so no large scale impacts are perfectly elastic.

Momentum is conserved in inelastic collisions, but one can't track kinetic energy through collision as some of it is converted to other forms of energy. Collisions in perfect gases approach perfectly elastic collisions, as do scattering interactions of sub-atomic particles that are deflected by electromagnetic force. Few large-scale interactions like slingshot type gravitational interactions among satellites and planets are perfectly elastic.

Equations for Kinetic Energy and Linear Momentum:

Let us examine the elastic collision between two particles of mass m1 and m2, respectively. Suppose that collision is head-on, so we are dealing with just one dimension. Velocities of particles before elastic collision are v1and v2, respectively. Velocities of particles after elastic collision are v'1and v'2. Applying the law of conservation of kinetic energy, we find:

1/2m1v12 + 1/2m2v22 = 1/2m1v'12 + 1/2m2v'22

Applying the law of conservation of linear momentum:

m1v1 + m2v2 = m1v'1 + m2v'2

These two equations put together will assist to solve any problem comprising elastic collisions. Generally, you will be provided quantities for m1, m2, and v1 and v2, can then manipulate two equations to solve for v1' and v'2.

Energy lost in perfectly inelastic collisions:

In any given system, total energy is usually the sum of numerous different forms of energy. Kinetic energy, KE, is the form related with motion and for single particle is written as:

KE-mv2/2

In contrast to momentum, kinetic energy is not a vector. For the system of many particles the total kinetic energy is simply sum of the individual kinetic energies of each particle:

KE-KE1+KE2+...

In an inelastic collision, the bodies collide and come apart again, but some kinetic energy is lost. That is, some kinetic energy is converted to another form of energy. The example would be collision between the baseball and bat where some kinetic energy is utilized to distort ball and converted in heat. If bodies collide and stick together, collision is known as perfectly inelastic. In this case, much of the kinetic energy is lost in collision. I.e., much of the kinetic energy is converted to other forms of energy.

Let us consider what kinetic energy must be in initial state before carts have hit each other. Using Equations, the initial kinetic energy KEi is

KEi = m1v1i2/2 + m2v2i2/2 -m1v1i2/2

While, final kinetic energy, after carts have hit and stuck together, is given by:

KEf = (m1 +m2)v2f/2

Notice that as carts are now stuck together mass is their total mass (m1+m2 )and they have a common velocity, vf-v1f-v2f.

For an inelastic collision, kinetic energy is not conserved but momentum IS. Using conservation of momentum (p→i-p→f) and the fact that Cart2 is initially at rest gives:

m1v→1i+m2v→2i-m1v→1i-(m1+m2)v →f

Using Eqs. we arrive at an equation for KEf in terms of KEi:

KEf-(m1/m1+m2)KEi

This is a prediction for the final kinetic energy of the perfectly inelastic collision.

Explosions:

Let us consider a case where two objects approach each other and merge in a frame of reference where the total momentum is zero. We also assume that these objects remain at rest after merging. When the opposite of this action occurs, that is, when an object at rest in such a frame of reference breaks up into two or more objects with an attendant sound, it becomes an explosion. The initial object of mass at rest breaks up into two objects, and they move with velocities such that the momentum is zero. That is their

m1v1+m2v2 = 0

From the law of energy conservation, once an object has initial potential energy U, then explosion is possible.

Therefore U = 1/2m1v12 + 1/2m2v22

Explosives utilized during wars have potential energy stored in molecules. When explosives are detonated, there is great release of energy.

Elastic and Inelastic Collisions:

In physics, collisions can be stated as either elastic or inelastic. When bodies collide in real world, they at times squash and deform to some degree. Energy to carry out deformation comes from objects' original kinetic energy. In other cases, friction turns some kinetic energy in heat. Physicists categorize collisions in closed systems (where net forces add together to zero) based on if colliding objects lose kinetic energy to some other form of energy:

Elastic collision: In the elastic collision, total kinetic energy in system is the same before and after the collision. If losses to heat and deformation are much smaller than other energies engaged, like when two pool balls collide and go their separate ways, you can usually ignore losses and say that kinetic energy was conserved.

Inelastic collision: In the inelastic collision, collision changes total kinetic energy in the closed system. In this situation, friction, deformation, or some other procedure transforms kinetic energy. If you can observe considerable energy losses because of non-conservative forces (like friction), kinetic energy is not conserved.

Elastic Collision Formula:

One dimensional Newtonian equation for elastic collision:

Consider two objects with mass m1 and m2.

Initial velocity of object 1 = u1,

Initial velocity of object 2 = u2

Let v1 and v2 be final velocities respectively.

Applying conservation of momentum principle, we get:

m1 u1 + m2 u2 = m1 v1 + m2 v2..................... (1)

Also applying principle of kinetic energy conservation:

12 m1 u12 + 12 m2 u22 = 12 m1 v12+ 12 m2 v22..... (2)

Solving the above equations for v1 and v2 we get:

v1 = u1 (m1-m2) + 2m2u2/ (m1+m2)..................... (3)

Similarly

v2 = u2 (m2-m1) + 2m1u1/(m1+m2).....................(4)

Also considering the trivial case such that there has been no collision yet:

v1 = u1 and v2 = u2................................ (5)

Yet another property: v1 - v2 = u2 - u1.

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