COLLISIONSL7/AS 9thJuly,09 7.30p.m. : COLLISIONSL7/AS 9thJuly,09 7.30p.m. Understanding collisions
Elastic & InElastic collisions.
Coefficient of Restitution.
Elastic collisions in
1& 2 –Dimensions
and their applications LECTURE BY PROF. ARDAMAN SIDHU hksidhuinstitute@gmail.com
Slide 2 : PROF. ARDAMAN SIDHU
HKSIS
PRESENTS
LECTURES IN PHYSICS
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COLLISIONS : COLLISIONS To the average person in the street the term collision is likely to mean some sort of automotive disaster. We'll use it in that sense, but we'll also broaden the meaning to include any strong interaction between bodies that lasts a relative short time. So we cover not only car accidents but also balls colliding on a biliard table, etc…. Cosmic collision
COLLISIONS : COLLISIONS DEFINATION
We define a collision as an isolated event in which two or more colliding bodies exert relatively strong forces on each other for a relatively short time.
COLLISIONS : COLLISIONS Thus actual physical contact between two bodies is not necessary for a collision. For example, an alpha particle speeding towards nucleus of an atom gets deflected by the electrostatic force of repulsion. Rutherford Gold Foil Experiment
Two key rules of the collision : Two key rules of the collision Two key rules of the collision game are:
Law of conservation of linear momentum, and
Two key rules of the collision : Two key rules of the collision Law of conservation of energy
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TYPES OF COLLISIONS : TYPES OF COLLISIONS Collisions between particles have been divided broadly into two types:
1. Elastic collision. 2. Inelastic collisions
3. Perfectly inelastic collisions.
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Elastic collision : Elastic collision A collision in which there is absolutely no loss of kinetic energy is called an elastic collision. For example, collision between atomic and sub atomic particles are elastic collisions.
BASIC CHARACTERISTICS OF AN ELASTIC COLLISION : BASIC CHARACTERISTICS OF AN ELASTIC COLLISION The linear momentum is conserved.
Total energy of the system is conserved.
The kinetic energy is conserved.
The forces involved during elastic collisions must be conservative forces.
INELASTIC COLLISION : INELASTIC COLLISION Inelastic collision
A collision in which there occurs some loss of kinetic energy is called an inelastic collision.
Therefore, collisions we come across in daily life are generally inelastic.
BASIC CHARACTERISTICS OF AN INELASTIC COLLISION : BASIC CHARACTERISTICS OF AN INELASTIC COLLISION The linear momentum is conserved.
Total energy of the system is conserved.
The kinetic energy is not conserved. Some part of Initial kinetic energy is converted in heat and sound energy.
The forces involved during inelastic collisions must be non-conservative forces.
PERFECTLY INELASTIC COLLISION : PERFECTLY INELASTIC COLLISION If two bodies stick to each other, after colliding the collision is said to be perfectly inelastic.
For example, mud thrown on a wall sticks to the wall. The loss of K.E. of mud is complete. The collision is perfectly inelastic.
Similarly, when an arrow gets stuck in a target and the two move together, the collision is perfectly inelastic.
Likewise, if a meteorite collides head on with earth, it becomes buried in earth. The collision is perfectly inelastic.
BASIC CHARACTERISTICS OF A PERFECTLY INELASTIC COLLISION : BASIC CHARACTERISTICS OF A PERFECTLY INELASTIC COLLISION The linear momentum is conserved.
Total energy is conserved.
Kinetic energy is NOT conserved. There is complete loss of kinetic energy as bodies stick together after collision.
Some or all the forces involved in an inelastic collision may be non-conservative in nature.
COEFFICIENT OF RESTITUTION OR COEFFICIENT OF RESILIENCE : COEFFICIENT OF RESTITUTION OR COEFFICIENT OF RESILIENCE It is defined as the ratio of relative velocity of separation after collision to the relative velocity of approach before collision. It is represented by ‘e’
where u1, u2 are velocities of two bodies before collision, and v1, v2 are their respective velocities after collision.
COEFFICIENT OF RESTITUTION OR COEFFICIENT OF RESILIENCE : COEFFICIENT OF RESTITUTION OR COEFFICIENT OF RESILIENCE For a perfectly elastic collision, relative velocity of separation after collision is equal to relative velocity of approach before collision. e = 1
For a perfectly inelastic collision, rel. vel. Of separation after collision e = 0
For all other collisions, e lies between 0 and 1 i.e. 0 < e < 1.
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION It involves two bodies moving initially along the same straight line, striking against each other without loss of kinetic energy and continuing to move along the same straight line after collision.
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION Suppose two balls A and B of masses m1 and m2 are moving initially along the same straight line with velocities u1 and u2, respectively, Fig. Below.
When u1 and u2, relative velocity of approach before collision,
= u1 – u2 …(1)
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION Hence the two balls collide. Let the collision be perfectly elastic. After collision, suppose v1 is velocity of A and v2 is velocity of B along the same straight line.
When v2 > v1, the bodies separate after collision.
= v2 – v1 …(2)
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION Linear momentum of the two balls before collision
= m1 u1 + m2 u2
Linear momentum of the two balls after collision
= m1 v1 + m2 v2
As linear momentum is conserved in an elastic collision therefore
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION Total K.E. of the two balls before collision
…….5
Total K.E. of the two balls after collision
……..6
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION As K.E. is also conserved in an elastic collision, therefore from (5) & (6)
…..(7)
Dividing (7) by (4), we get
or v2+u2 = u1 + v1 …..(8)
ELASTIC COLLISION IN ONE DIMENSION : ELASTIC COLLISION IN ONE DIMENSION Hence in one dimensional elastic collision, relative velocity of separation collision is equal to relative velocity of approach before collision
Hence coefficient of restitution/resilience of a perfectly elastic collision in one dimension is unity.
Calculation of velocities after collision : : Calculation of velocities after collision : Velocity of A:
From (8) ….(9)
Putting in (3), we get
…(10)
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Calculation of velocities after collision : : Calculation of velocities after collision : Velocity of B:
Put this value of v1 from (10) in (9)
(11)
Particular cases :ELASTIC COLLISION IN 1-D : Particular cases :ELASTIC COLLISION IN 1-D 1. When masses of two bodies are equal i.e.
From (10),
From (11),
i.e. velocity of A after collision = vel. Of B before collision.
Calculation of velocities after collision : : Calculation of velocities after collision : When masses of two bodies are equal
i.e. velocity of B after collision = velocity of A before collision.
Hence when two bodies of equal masses undergo elastic collision in one dimension, their velocities are just interchanged.
This result has an important application in a nuclear reactor. Fast moving neutrons in a nuclear reactor are slowed down by making them collide against the nuclei of moderator.
Calculation of velocities after collision : Calculation of velocities after collision 2. When the target body B is initially at rest
From (10), …(12)
From (11), …(13)
These cases arise further:
Calculation of velocities after collision : : Calculation of velocities after collision : 2. When the target body B is initially at rest
2(1) When masses of two bodies are equal
i.e. m1 = m2
From (12), v1 = 0
From (13)
i.e. body A comes to rest and body B starts moving with the initial velocity of A.
Calculation of velocities after collision : : Calculation of velocities after collision : 2. When the target body B is initially at rest
2(2)When body B at rest is very heavy
i.e. m2 > > m1 i.e. m1 can be ignored compared to m2
Putting m1 = 0 in (12) and (13), we obtain
Hence when a light body A collides against a heavy body B at rest : A rebounds with its own velocity and B continues to be at rest.
Calculation of velocities after collision : : Calculation of velocities after collision : When body B at rest has negligible mass
i.e. m2 > > m1 i.e. m2 can be ignored compared to m1
Putting m2 = 0 in (50) and (51), we get
Hence when a heavy body A undergoes an elastic collision with a light body B at rest, the body A keeps on moving with the same velocity of its own and the body B starts moving with double the initial velocity of A.
INELASTIC COLLISION IN ONE DIMENSION : INELASTIC COLLISION IN ONE DIMENSION Two bodies of masses m1and m2 moving with velocities u1 and u2 respectively, along a single axis. They collide involving some loss of kinetic energy. Therefore, the collision is inelastic. Let v1 and v2 be the velocities of the two bodies after collision.
We can write the law of conservation of linear momentum for the two body system as:
…(14)
Perfectly inelastic collision in one dimension : Perfectly inelastic collision in one dimension Two bodies of mass m1 and m2 the body of mass m2 happens to be initially at rest (u2 = 0 ).
We refer to this body as the target.
The incoming body of mass m1, moving with initial velocity u1 is referred to as the projectile.
After the collision, the two bodies move together with a common velocity V. The collision is perfectly inelastic.
Perfectly inelastic collision in one dimension : Perfectly inelastic collision in one dimension As the total linear momentum of the system cannot change, therefore Pt = Pf
…(53)
Perfectly inelastic collision in one dimension : Perfectly inelastic collision in one dimension Total K.E. before collision =
Total K.E. after collision =
=
Loss of K.E. =
which is positive
Therefore, some K.E. is always lost in an inelastic collision.
ELASTIC COLLISION IN TWO DIMENSIONS OR OBLIQUE COLLISION : ELASTIC COLLISION IN TWO DIMENSIONS OR OBLIQUE COLLISION In general, the collisions are two dimensional, where the initial velocities and the final velocities may lie in a plane.
So, two bodies traveling initially along the same straight line collide without loss of kinetic energy and move along different directions in a plane, after collision, the collision is said to be elastic collision in two dimensions.
ELASTIC COLLISION IN TWO DIMENSIONS OR OBLIQUE COLLISION : ELASTIC COLLISION IN TWO DIMENSIONS OR OBLIQUE COLLISION Suppose m1, m2 are the masses of two bodies A and B moving initially along X-axis with velocities u1 and u2 respectively.
When u1 > u2, the two bodies collide. After collision, let the body A move with a velocity v1 at an angle ? with X-axis. Let the body B move with a velocity v2 at an angle ? with X-axis as shown in fig.
As the collision is elastic, kinetic energy is conserved.
? Total K.E. After collision
= Total K.E. Before collision.
ELASTIC COLLISION IN TWO DIMENSIONS OR OBLIQUE COLLISION : ELASTIC COLLISION IN TWO DIMENSIONS OR OBLIQUE COLLISION As linear momentum is conserved in elastic collision, therefore, along X-axis, total linear momentum after collision = total linear momentum before collision.
Now, along Y-axis, linear momentum before collision is zero (as both the bodies are moving along X-axis)
?
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