Collisions 136212925

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```					Collisions

A collision is an event in which two or more bodies interact, for a relatively short time,
so that there is exchange of momentum and energy between them. The event could be
such that the bodies involved come into physical contact with one another, such as the
collisions between two balls or between a car and a truck. Or, the event could be such
that the bodies interact from a distance via a field like the electromagnetic or gravitational
field. These interactions are also called collisions although bodies involved do not come
into contact with one another. All collisions between atomic particles are caused by the
electromagnetic field. A well-known example is the scattering of  particles by the
nuclei.

α particle

Nucleus
Fig. 9.1. A beam
of α particles is
incident on a
sheet of metal.
Most particles
pass through
undeflected.
Some get
deflected while a
on collisions with
the nuclei.

For an animation of Rutherford scattering, visit:
webphysics.Davidson.edu/Applets/pqp_preview/contents/pqp_errata/

The interaction of a planet or a comet with the sun is an example of a collision that is
caused by the intervention of the gravitational field. Other examples are (i) the slingshot
effect (or gravity assist) in which a satellite is made to pass close to a planet to gain
energy, and (ii) the collision between two galaxies.
Fig. 9.2. Collision of two galaxies (NASA/Hubble
Heritage Team (STScI).

Fig. 9.3. This sketch shows how
a satellite launched from the
earth can be made to gain energy
by making it pass close to a
planet like Venus.

If the system of colliding particles is isolated, then their total momentum is conserved,
that is, the total momentum before the collision is the same as after the collision.

Before                                              After

Fig. 9.4. The momentum of the railcars before and after coupling must be the same.

       
For simplicity let us consider only two-particle collision. If p1i and p 2i are the initial
         
momenta and p1 f and p 2 f are the final momenta of the two particles, then we must have
for momentum conservation (sub-indices 1 and 2 are used for the two particles)
                 
p1i  p 2i  p1 f  p 2 f .                      (9.1)
If particles have velocities small compared with the velocity of light (always true for
     
ordinary bodies like balls and trucks) we can write p  mv . Then Equation (9.1) can be
written as
                         
m1v1i  m2 v 2i  m1v1 f  m2 v 2 f ,                (9.2)

where m1 and m 2 are the masses of the two particles. Notice that this last equation is a
vector equation. In three dimensions, this is equivalent to three scalar equations. In two
dimensions, there are two scalar equations. In one dimension, there is just one scalar
equation. This means that in one dimension we need to consider only the magnitudes
of the velocities.

Collisions are of two types. A collision in which the linear momentum and kinetic
energy are conserved is called an elastic collision. When momentum is conserved but
the kinetic energy is not (but the total energy is) conserved, the collision is said to be an
inelastic collision.

Collisions between ordinary objects, such as balls or cars or trucks, are all inelastic to
some extent because a fraction of kinetic energy is dissipated as heat, light, sound or
some other form, but for ease of calculation, we consider them as elastic. Collisions
between hard steel balls are very nearly elastic. Collisions between molecules of an ideal
gas are also considered elastic to make it easier to calculate the pressure exerted by the
gas. Scattering mediated by electromagnetic fields or large scale interactions involving
gravitational fields are perfectly elastic.

A good example of an elastic collision is the bouncing of a hard ball from a smooth hard
floor (marble). The ball bouces back to the same height every time, indicating no loss in
kinetic energy. If, however, the floor is not hard (say, it is carpeted) or the ball is not
hard, then the ball does not rise to the same height, indicating a loss in kinetic energy.
The collisons of this kind are examples of inelastic collisions.

Fig. 9.5. Examples of elastic (left) and inelastic (right) collisions. In the former case, the
ball attains the same height after bouncing from the floor. In the latter case, it does not.
Now suppose that a collision is elastic. Then the conservation of kinetic energy implies
 2       2      2        2
m1v1i  m2 v2i  m1v1 f  m2 v2 f .                                      (9.3)

Equations (9.1) to (9.3) allow us to relate the final velocities of the colliding particles
with their initial velocities. Equation (9.3) can be rewritten as
                                                    
m2 (v 2 f  v 2i )( v 2 f  v 2i )  m1 (v1 f  v1i )( v1 f  v1i ) ,   (9.4)

which, with the use of Equation (9.2), can be written as
                       
(v 2 f  v1 f )  (v 2i  v1i ) .                                       (9.5)

The term on the left hand side, the relative final velocity of the two participating
particles, is called the velocity of recession, while that on the right is called the velocity
of approach, the relative initial velocity of the two particles. So, the velocity of
recession is equal in magnitude to the velocity of approach but is in the opposite
direction. In fact the ratio

,                                                      (9.5 a)

called the coefficient of restitution is used as a measure of how elastic a collision is.
Following from Equation (9.5) it is equal to 1 for perfectly elastic collisions. For
perfectly inelastic collisions, e = 0. For partly elastic collisions, e can have values
anywhere between 0 and 1.

Fig. 9.6. In the top row, the particles
approach each other. In the central
row, they collide. In the third row
they recede from each other. If the
collision is elastic, then the velocity
of recession must be equal to the
velocity of approach.

A simple case is that of a collision in which the colliding particles stick together and

move finally with a common velocity v f . Then, from conservation of momentum alone
we get
        
    m v  m2 v 2i
v f  1 1i         .                                     (9.6)
m1  m 2

To make calculations simpler, it is always possible to consider the second particle, also
called the target, to be initially at rest. In that case,

     m1v1i
vf           .                                         (9.7)
m1  m 2

If the particles have equal masses, then the final velocity of the two sticking together
is half the initial velocity of the incident particle. This is also what common sense
would dictate.

Fig. 9.7. Two particles of
equal mass collide and stick
together. Conservation of
momentum implies that the
final velocity of the couple is                                         Target
half the initial velocity of the
approaching particle.

Let us now look at general collisions in one dimension. Remember that in Equation (9.2)
we need to consider only magnitudes. Then Equations (9.2) and (9.3) can be solved to
give

m1  m2        2m 2
v1 f            v1i          v 2i ,                     (9.8)
m1  m2       m1  m2

2m1          m  m2
v2 f             v1i  1       v 2i .                    (9.9)
m1  m 2      m1  m 2

Let us suppose that the target particle was at rest before the collision ( v 2i  0) . In fact, it
is always possible to work in a frame of reference in which the target is at rest and
other velocities are measured with respect to this frame. In that case, we have
m1  m 2
v1 f             v1i ,                                (9.10)
m1  m 2

2m1
v2 f              v1i .                                 (9.11)
m1  m 2

Depending on the relative magnitudes of the masses of the two particles, we have the
following three interesting cases:

Case 1: Masses of the two particles are equal ( m1  m2 ). From Equations (9.10) and
(9.11) we get

v1 f  0, and v 2 f  v1i .                            (9.12)

Before                           After

Fig. 9.8. Blue and yellow balls have equal mass. The yellow ball is initially at rest.
After collision the blue ball comes to rest while the yellow ball starts moving with the
velocity that the blue ball had initially.

So, in a collision of two particles of equal mass in which the target particle is at rest, the
incident particle comes to rest after the collision and transfers all its velocity to the target
particle. [For illustration, visit: solomom.physics.sc.edu/~tedeschi/demo/movies/

You must have seen, or even played with, a toy called Newton’s Cradle. It consists of
several identical pendulums with steel bobs (5 or 7 in most cases) hung side-by-side, or
several steel balls fitted in a groove.

Fig. 9.9. A
If you move one ball a little aside and let it strike with the remaining balls, one ball at the
end is seen to move the same distance and with the same velocity as the first ball, as
shown in Fig. 9.9. Since the balls are identical and collision is elastic, momentum and
energy equal to the momentum and energy of the first ball is passed from one ball to the
next, till it reaches the last ball. The movement of the last ball with the same velocity and
by the same distance as the velocity and distance of the first ball shows the conservation
of momentum and kinetic energy in a collision between identical balls. For an animation
www.physics.ucla.edu/demoweb/demomanual/mechanics/momentum_and_collisions/col
lision_balls.html

We see from Equation (9.12) that the incident particle transfers all its kinetic energy to
the target particle. Even if the masses of the colliding particles are not exactly equal,
sharing of energy between the particles during collisions is quite effective. This fact
forms the basis of the use of moderators in nuclear reactors. The neutrons that are
released in the fission process are very energetic and the probability of their causing
further fission is low. They must be slowed down to thermal energies so that their
probability of causing fission increases. The fast neutrons are made to lose energy by
making them collide with the molecules of a moderator. The moderator is a light
material such as ordinary water, heavy water or graphite, so that there is an effective
sharing of energy.

Case 2: Mass of the incident particle is much smaller than the mass of the target
particle ( m1  m2 ). In this case we have

2m1
v1 f  v1i , and v 2 f        v1i  v1i .           (9.13)
m2

The implication is that the lighter particle rebounds with the same velocity as it had while
going forward, while the heavy target particle is almost unaffected. Accordingly, when a
light car collides with a heavy truck, the car is thrown back with the speed it came with
while the truck hardly budges from its place.

For illustration of these cases, visit: hyperphysics.phys-astr.gsu.edu/hbase/colsta.html,
and for animations involving collisions between a car and a truck, visit
www.glenbrook.k12.il.us/gbssci/phys/mmedia/momentum/cthoe.html.
Fig. 9.10. A small
truck crashes against
a bigger truck at rest.
It comes back with
the speed with which
it was moving when
it struck the larger
truck.

An important application of this case is the use of gravity assist phenomenon to
accelerate a spacecraft by making it pass close to a planet. You remember that in an
isolated system a collision assisted by the gravitational field is elastic. Look at the
illustration in Fig. 9.11 below to see how gravitational field of a large planet can cause a
change in the orbit of a comet.

Let us now see how gravity assist works. We model the interaction between the
spacecraft and the planet as a head – on elastic collision. Imagine that in the frame of
Original comet orbit
Jupiter’s orbit

Jupiter

SUN

Comet orbit modified by Jupiter

Fig. 9.11. The change in the orbit of a long – period comet as it passes close to the giant
planet Jupiter. The modified orbit makes the comet a short – period comet.

reference of the sun (in which the sun is at rest) the planet is moving towards left with
speed U. A spacecraft approaches the planet with a speed u moving to the right. An
observer on the planet, that is in the frame of reference of the planet, would see the
spacecraft approach it with speed (U + u). Since the mass of spacecraft is much smaller
than the mass of the planet, Equation (9.13) would apply. So, the observer on the planet
would see the spacecraft rebounding to the left with a speed of (U + u). The planet, by
virtue of being much more massive, is not affected. Its own orbit is not disturbed. Let us
move back now to the frame of reference of the sun. In this frame the planet is moving to
the left with velocity U. This velocity must, therefore, be added to the velocity of the
spacecraft. Therefore, in the frame of the sun, the frame with which we are really
concerned, the spacecraft moves to the left with velocity (u + U) + U = u + 2U.
Depending on the magnitudes of u and U, the impact can give a substantial boost to the

U
u
U
Spacecraft                                           Planet
u

In Sun’s Frame of Reference – Before Collision

u+U

Spacecraft                                           Planet            Before Collision

u+U

Spacecraft                                           Planet           After Collision

In planet’s Frame of Reference

u + 2U                         U                u + 2U

Spacecraft                                           Planet                  U

In Sun’s Frame of Reference – After Collision

Fig. 9.12. A simplified head – on collision between a spacecraft and a planet. The
spacecraft is reflected with increased energy. Actually, there is no head-on collision. The
spacecraft goes round the planet.
energy of the spacecraft, helping it to conserve its precious fuel. The effect is also called
the slingshot effect. If the spacecraft interacts with more than one planets, as was
designed to happen with the spacecraft Cassini, it gains energy at encounter with each
planet. The net result is a substantial gain in energy of the spacecraft. For an animation
of gravity assist, visit: http://www.esa.int/esaSC/SEMXLE0P4HD_index_0.html

It is important to remember that the timing of the maneuver is important. At another
time, when the collision is not head-on, the spacecraft could be decelerated.
Sometime, simply a change in the orbit of the spacecraft may be desired. So,
depending on their requirement, the scientists have to plan the maneuver very carefully.

Fig. 9.13. The trajectory of the
spacecraft Cassini. Notice that the
trajectory passes close to several
planets to get boost in energy.
Source:
/ast18aug99_1.htm

Case 3: Mass of the incident particle is much larger than the mass of the target
particle ( m1  m2 ). In this case the general formulae (9.10) and (9.11) give

v1 f  v1i , and v 2 f  2v1i .                          (9.14)

We see that there is no change in the velocity of the incident particle, while the target is
forced to move in the same direction as the incident particle with twice the velocity of the
incident particle (for animation, visit: physics-
animations.com/Physics/English/par_txt.htm). It is because of this that a heavy truck
hitting a light car from behind can send the car hurtling down with a huge speed, while it
itself keeps moving with its original speed.
U

Fig. 9.14. Collision
of a truck with a
stationary car. The
car hurtles down
U
with speed 2U.

2U

Consider once again the collision between two particles. Notice that in general the
collision is in a plane, because the cooliding particles do not always collide head – on.
The plane is defined by the relative velocity of the colliding particles and the directions
of the force between them, which is generally along the line joining the two particles. The
velocity vectors before and after the collision are shown (Fig. 9.15). Remember that
Equation (9.5) demands that the velocity of approach is equal to the velocity of recession.
This is shown by the traiangle formed by the vector velocities.

Fig. 9.15.
Elastic collison
between two
particles in two
dimensions.
Notice that the
sum of the final
momentum
vectors is equal
to the initial         Before collision          After collision
momentum
vector of particle
1.

In the special case where the two particles have equal masses and one of them is at rest,
being the target, we have from conservation of momentum
           
v1i  v1 f  v 2 f ,                            (9.15)
as shown in the vector diagram in Fig. 9.15. On the other hand, from the conservation of
kinetic energy we get
 2  2  2
v1i  v1 f  v2 f ,                            (9.16)

Triangle of vectors
Before collision              After collision

Fig. 9.16. Elastic collision of two particles of equal mass.

which shows that the vector triangle is a right-angled triangle. This implies that the final
velocities of the two particles are always at right angles to each other. On a billiard table,
if a moving ball strikes a stationary ball, the two balls move at right angles to each other.
The players use this fact to their advantage and try to score maximum points. For an
animation of a two-dimensional collision, visit:en.wikipedia.org/wiki/Elastic_collision

Fig. 9.17. On a billiard
table, the two balls move
at right angles to each
other after they strike.

Two-body collisions appear particularly simple when viewed in a frame of reference in
which the centre of mass of the two particles is at rest. Such a frame is called the centre
of mass frame. If the system is isolated, that is, there is no external force acting on the
system, the centre of mass is always at rest. The linear momentum of the two particles is
zero initially as well as finally. In this frame, if a particle approaches the other with
momentum , the latter must approach the former with momentum                , so that the net
linear momentum is zero. After collision, too, the two particles are seen moving away
from each other with equal momentum.
Fig. 9.18. Two-particle
collision seen from the
centre of mass frame of
reference.

Collisions play an important role in determining the physical state of a system of
particles, say a gas. If we assume the collisions between the particles of a perfect gas to
be elastic, then collisions between the particles bring about sharing of kinetic energy
among them. If the density is high, the sharing of energy is efficient. A thermal
equilibrium is established in the system. The velocities of gas particles approach a
distribution called the Boltzmann – Maxwell distribution. Such a distribution is
indicative of a thermal equilibrium. A single temperature and pressure can be defined
for such a system. If the density is low, then the collisions are infrequent and system
does not attain a thermal equilibrium. No single temperature can be defined for the
system. The particles of such a system do not obey Boltzmann – Maxwell distribution of
velocities. (See Temperature and Thermometer.) [For an animation of the equilibrium
velocity distribution, visit: jersey.uoregon.edu/Balloon/]

The Maxwell – Boltzmann distribution is given by
,                       (9.17)
where        is the number of particles with velocity , N is the total number of particle in
the system, k is the Boltzmann constant and T is the temperature of the system.

Fig. 9.19. Boltzmann – Maxwell
distribution shows the number of particles
of various speeds at a given temperature.
A gas in equilibrium obeys such a              No. of
distribution. The red curve is for a lower     particles
temperature and the blue curve is for a
higher temperature. Notice that as
temperature increases the peak speed of
the particles also increases
Speed

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