System of particles

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Session
System of Particle - 4
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Session Objectives

1. Collision-elastic and inelastic
2. Elastic collisions in one and two dimension
3. Coefficient of restitution
4. Explosions
5. Explosion in a projectile

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Collision between two particles

• Since Fext0 so PiPf

• if KEiKEf Perfectly Elastic

• if KEi gt KEf Inelastic

All elastic collisions are perfectly elastic
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Elastic collisons

• Since PiPf

• Since KEiKEf

NOTE Solving these two equations you can get the
values of final velocities
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Inelastic collisons

• Since PiPf

• Since KEi gtKEf

NOTE You need one more equality to solve for
final velocities.
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Coefficient of restitution
Velocity of separation e (velocity of approach)
When e1 elastic collision e0 completely
inelastic collision 0 ? e ? 1Inelastic collision
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General speed relations (Remember)
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Final speeds (special cases Elastics collision)
(a) Heavy body and light body
(b) If m2gtgtm1(a light body hits a heavy body
from behind)
(c) Bodies of equal mass (m1m2)
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Illustrative problem
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Solution
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Special case perfectly inelastic collision (e0)
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Two Dimensional Collisions
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Elastic collision in 2D
Conservation of momentum
NOTE There are 3 equations but 4 variables. So
one physical condition is required to solve the
problem.
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Illustrative Problem
A ball of mass 1kg moving with velocity V5i
collides with another ball of mass 2 kg
initially at rest and at origin. The first ball
comes to rest after collision and second breaks
into two equal pieces. one piece starts moving
with a velocity V3j m/s. Then the velocity of
other piece is
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Solution
In x-direction
1x42x0 1x v cos?
In y-direction
0 1x3 1xvsin?
Using given equation
tan?3/4 and v 5m/s
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Class Exercise - 1
A light spring of spring constant K 1,000 N/m
is kept compressed between two blocks of masses m
10 kg and M 15 kg on a smooth horizontal
surface. When released the blocks acquire
velocities in opposite directions. The spring
loses contact with the blocks when it acquires
natural length. If the spring was initially
compressed through a distance x 15 m, find the
final speeds of the two blocks.
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Solution
When spring acquired its natural length, let the
velocities of two blocks be v and u respectively.
As no external force acts on the system,
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Solution
1.16 m/s
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Class Exercise - 2
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Solution
Velocity of B before collision 0
Velocity of A after collision v
Velocity of B after collision v'
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Solution
Then v v' e(u)
Let t' be the time after next impact takes place,
then
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Class Exercise - 3
A neutron travelling with a velocity v and
kinetic energy E collides elastically head-on
with the nucleus of an atom of mass number A at
rest. What fraction of total energy is retained
by the neutron?
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Solution
1 V 1V1 AVA
Þ V2 V12 AVA2
Using (i),
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Class Exercise - 4
A moving body with a mass m1 strikes a stationary
body of mass m2. The masses m1 and m2 should be
in the ratio m1/m2 so as to decrease the velocity
of the first body 0.5 times assuming a perfectly
elastic impact. What is the ratio of m1/m2 ?
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Solution
Applying momentum conservation,
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Solution
Also,
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Class Exercise - 5
A ball of mass m moving with a speed u undergoes
a head on elastic collision with a ball of mass
nm initially at rest. What fraction of the
incident energy is transferred to the heavier
ball?
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Solution
Applying momentum conservation,
Also, v2 v1 u
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Class Exercise - 6
A body of mass m moving with a velocity v1 in the
x-direction collides with another body of mass M
moving in y-direction with a velocity v2. They
coalesce into one body during collision. What is
the direction and magnitude of the momentum of
the final body?
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Solution
From conservation of momentum in x- and
y-directions, we get
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Class Exercise - 7
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Solution
Let A be projected with a velocity u.
Using (i) and (ii), we get
Now, ball A moves towards the wall and returns
back, say with a velocity VA, then
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Solution
Solving (iii) and (iv), we get
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Class Exercise - 8
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Solution
Along X-axis,
mu cosa mv1x mv2x
When maximum deformation takes place,
v1x v2x
u cosa v1x v2x 2v1x
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Solution
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Class Exercise - 9
A ball moving with a speed of 9 m/s strikes an
identical stationary ball such that after the
collision, the direction of each ball makes an
angle of 30 with the original line of motion.
Find the speeds of the two balls after the
collision if the kinetic energy is conserved in
the collision process.
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Solution
Applying momentum of conservation in x- and
y-directions, we get
Also, in y-direction,
Using equations (i) and (ii), we get
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Solution
Kinetic energy is not conserved.
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Class Exercise - 10
Two blocks of masses m1 2 kg and m2 5 kg are
moving in the same direction along a frictionless
surface with speeds 10 m/s and 3 m/s
respectively, m2 being ahead of m1. An ideal
spring with K 1,120 N/m is attached to the back
side of m2. Find the maximum compression of the
spring when the blocks collide.
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Solution
Let V be the velocity of the blocks of the time
of maximum compression. Note that both the blocks
will move with equal speeds at the time of
maximum compression. Hence,
From energy conservation,
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Solution
? 100 65 560x2
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