Short Notes: Collisions and Virtual Work | Short Notes for Mechanical Engineering PDF Download

 Page 1


Collisions and Virtual Work
A collision occurs when two or more objects hit each other. When objects collide, 
each object feels a force for a short amount of time. This force imparts an impulse 
or changes the momentum of each of the colliding objects. But if the system of 
particles is isolated, we know that momentum is conserved. Therefore, while the 
momentum of each individual particle involved in the collision changes, the total 
momentum of the system remains constant.
The collision between two bodies may be classified in two ways: Head-on collision, 
and Oblique collision.
• Head-on Collision
° Let the two balls of masses mi and m2 collide directly with each other 
with velocities Vi and V 2 in the direction as shown in figure.
° After the collision, the velocity becomes and along the same line.
• Oblique Collision
° In case of oblique collision linear momentum of individual particle do 
change along the common normal direction. No component of impulse 
act along the common tangent direction. 
o So, linear momentum or linear velocity remains unchanged along 
tangential direction. Net momentum of both the particle remains
Before collision After collision
Page 2


Collisions and Virtual Work
A collision occurs when two or more objects hit each other. When objects collide, 
each object feels a force for a short amount of time. This force imparts an impulse 
or changes the momentum of each of the colliding objects. But if the system of 
particles is isolated, we know that momentum is conserved. Therefore, while the 
momentum of each individual particle involved in the collision changes, the total 
momentum of the system remains constant.
The collision between two bodies may be classified in two ways: Head-on collision, 
and Oblique collision.
• Head-on Collision
° Let the two balls of masses mi and m2 collide directly with each other 
with velocities Vi and V 2 in the direction as shown in figure.
° After the collision, the velocity becomes and along the same line.
• Oblique Collision
° In case of oblique collision linear momentum of individual particle do 
change along the common normal direction. No component of impulse 
act along the common tangent direction. 
o So, linear momentum or linear velocity remains unchanged along 
tangential direction. Net momentum of both the particle remains
Before collision After collision
conserved before and after collision in any direction.
Oblique collision
General Equation for Velocity after Collision
\ /
fflj - em2 m, 4- em,
m l — m 2 m l 4- m 2
m , — em: | m x + em2
+ m 2
*
[ + m 2 J
Where mi = mass of body 1
• m2 = mass of body 2
• vi = velocity of body 1
• v2 = velocity of body 2= velocity of body 1 after collision= velocity of body 2 
after collision
Where e = coefficient restitution
• In case of head-on elastic collision e = 1
• In case of head-on inelastic collision 0 < e < 1
• In case of head-on perfectly inelastic collision e = 0
The procedure for analyzing a collision depends on whether the process 
is elastic or inelastic. Kinetic energy is conserved in elastic collisions, whereas 
kinetic energy is converted into other forms of energy during an inelastic collision. 
In both types of collisions, momentum is conserved.
• Elastic Collisions
° Some kinetic energy is converted into sound energy when pool balls 
collide otherwise, the collision would be silent and a very small amount 
of kinetic energy is lost to friction.
° However, the dissipated energy is such a small fraction of the ball’s 
kinetic energy that we can treat the collision as elastic.
Equations for Kinetic Energy and Linear Momentum
• Let's examine an elastic collision between two particles of mass
Page 3


Collisions and Virtual Work
A collision occurs when two or more objects hit each other. When objects collide, 
each object feels a force for a short amount of time. This force imparts an impulse 
or changes the momentum of each of the colliding objects. But if the system of 
particles is isolated, we know that momentum is conserved. Therefore, while the 
momentum of each individual particle involved in the collision changes, the total 
momentum of the system remains constant.
The collision between two bodies may be classified in two ways: Head-on collision, 
and Oblique collision.
• Head-on Collision
° Let the two balls of masses mi and m2 collide directly with each other 
with velocities Vi and V 2 in the direction as shown in figure.
° After the collision, the velocity becomes and along the same line.
• Oblique Collision
° In case of oblique collision linear momentum of individual particle do 
change along the common normal direction. No component of impulse 
act along the common tangent direction. 
o So, linear momentum or linear velocity remains unchanged along 
tangential direction. Net momentum of both the particle remains
Before collision After collision
conserved before and after collision in any direction.
Oblique collision
General Equation for Velocity after Collision
\ /
fflj - em2 m, 4- em,
m l — m 2 m l 4- m 2
m , — em: | m x + em2
+ m 2
*
[ + m 2 J
Where mi = mass of body 1
• m2 = mass of body 2
• vi = velocity of body 1
• v2 = velocity of body 2= velocity of body 1 after collision= velocity of body 2 
after collision
Where e = coefficient restitution
• In case of head-on elastic collision e = 1
• In case of head-on inelastic collision 0 < e < 1
• In case of head-on perfectly inelastic collision e = 0
The procedure for analyzing a collision depends on whether the process 
is elastic or inelastic. Kinetic energy is conserved in elastic collisions, whereas 
kinetic energy is converted into other forms of energy during an inelastic collision. 
In both types of collisions, momentum is conserved.
• Elastic Collisions
° Some kinetic energy is converted into sound energy when pool balls 
collide otherwise, the collision would be silent and a very small amount 
of kinetic energy is lost to friction.
° However, the dissipated energy is such a small fraction of the ball’s 
kinetic energy that we can treat the collision as elastic.
Equations for Kinetic Energy and Linear Momentum
• Let's examine an elastic collision between two particles of mass
, respectively. Assume that the collision is head-on, so we are dealing with 
only one dimension— you are unlikely to find two-dimensional collisions of any 
complexity on SAT II Physics. The velocities of the particles before the elastic 
collision are
l’i
and
V ,
, respectively. The velocities of the particles after the elastic collision are
V
and
'Y
. Applying the law of conservation of kinetic energy, we find:
Applying the law of conservation of linear momentum:
m it» i + — miiV +
• These two equations put together will help you solve any problem involving 
elastic collisions. Usually, you will be given quantities for
mi
and
, and can then manipulate the two equations to solve for
V,’
and
»Y
• A head-on with the cue ball in pool, Both of these balls have the same mass, 
and the velocity of the cue ball is initially
. What are the velocities of the two balls after the collision? Assume the 
collision is perfectly elastic
Page 4


Collisions and Virtual Work
A collision occurs when two or more objects hit each other. When objects collide, 
each object feels a force for a short amount of time. This force imparts an impulse 
or changes the momentum of each of the colliding objects. But if the system of 
particles is isolated, we know that momentum is conserved. Therefore, while the 
momentum of each individual particle involved in the collision changes, the total 
momentum of the system remains constant.
The collision between two bodies may be classified in two ways: Head-on collision, 
and Oblique collision.
• Head-on Collision
° Let the two balls of masses mi and m2 collide directly with each other 
with velocities Vi and V 2 in the direction as shown in figure.
° After the collision, the velocity becomes and along the same line.
• Oblique Collision
° In case of oblique collision linear momentum of individual particle do 
change along the common normal direction. No component of impulse 
act along the common tangent direction. 
o So, linear momentum or linear velocity remains unchanged along 
tangential direction. Net momentum of both the particle remains
Before collision After collision
conserved before and after collision in any direction.
Oblique collision
General Equation for Velocity after Collision
\ /
fflj - em2 m, 4- em,
m l — m 2 m l 4- m 2
m , — em: | m x + em2
+ m 2
*
[ + m 2 J
Where mi = mass of body 1
• m2 = mass of body 2
• vi = velocity of body 1
• v2 = velocity of body 2= velocity of body 1 after collision= velocity of body 2 
after collision
Where e = coefficient restitution
• In case of head-on elastic collision e = 1
• In case of head-on inelastic collision 0 < e < 1
• In case of head-on perfectly inelastic collision e = 0
The procedure for analyzing a collision depends on whether the process 
is elastic or inelastic. Kinetic energy is conserved in elastic collisions, whereas 
kinetic energy is converted into other forms of energy during an inelastic collision. 
In both types of collisions, momentum is conserved.
• Elastic Collisions
° Some kinetic energy is converted into sound energy when pool balls 
collide otherwise, the collision would be silent and a very small amount 
of kinetic energy is lost to friction.
° However, the dissipated energy is such a small fraction of the ball’s 
kinetic energy that we can treat the collision as elastic.
Equations for Kinetic Energy and Linear Momentum
• Let's examine an elastic collision between two particles of mass
, respectively. Assume that the collision is head-on, so we are dealing with 
only one dimension— you are unlikely to find two-dimensional collisions of any 
complexity on SAT II Physics. The velocities of the particles before the elastic 
collision are
l’i
and
V ,
, respectively. The velocities of the particles after the elastic collision are
V
and
'Y
. Applying the law of conservation of kinetic energy, we find:
Applying the law of conservation of linear momentum:
m it» i + — miiV +
• These two equations put together will help you solve any problem involving 
elastic collisions. Usually, you will be given quantities for
mi
and
, and can then manipulate the two equations to solve for
V,’
and
»Y
• A head-on with the cue ball in pool, Both of these balls have the same mass, 
and the velocity of the cue ball is initially
. What are the velocities of the two balls after the collision? Assume the 
collision is perfectly elastic
L'sto re
a ft©
Substituting
m | = m j = m
and
v 2 = 0
into the equation for conservation of kinetic energy we find:
-mpj = -nj(t)!/2 + I'i?)
V ? = v f + Vi?
Applying the same substitutions to the equation for conservation of momentum, we 
find:
rnt'i = m v\l + m vjl 
V t = vt'+i V
If we square this second equation, we get:
t’| = vi? + V 2 > l + 2 i'|/r> 2 /
By subtracting the equation for kinetic energy from this equation, we get:
2i.'i/rV = 0
• The only way to account for this result is to conclude that
V\ - 0
and consequently.
vt = vi
Inelastic Collisions
• Most collisions are inelastic because kinetic energy is transferred to other 
forms of energy-such as thermal energy, potential energy, and sound-during 
the collision process.
• The kinetic energy is not conserved, in inelastic collision. Momentum is 
conserved in all inelastic collisions.
• The one exception to this rule is in the case of completely inelastic collisions.
• Completely Inelastic Collisions
Page 5


Collisions and Virtual Work
A collision occurs when two or more objects hit each other. When objects collide, 
each object feels a force for a short amount of time. This force imparts an impulse 
or changes the momentum of each of the colliding objects. But if the system of 
particles is isolated, we know that momentum is conserved. Therefore, while the 
momentum of each individual particle involved in the collision changes, the total 
momentum of the system remains constant.
The collision between two bodies may be classified in two ways: Head-on collision, 
and Oblique collision.
• Head-on Collision
° Let the two balls of masses mi and m2 collide directly with each other 
with velocities Vi and V 2 in the direction as shown in figure.
° After the collision, the velocity becomes and along the same line.
• Oblique Collision
° In case of oblique collision linear momentum of individual particle do 
change along the common normal direction. No component of impulse 
act along the common tangent direction. 
o So, linear momentum or linear velocity remains unchanged along 
tangential direction. Net momentum of both the particle remains
Before collision After collision
conserved before and after collision in any direction.
Oblique collision
General Equation for Velocity after Collision
\ /
fflj - em2 m, 4- em,
m l — m 2 m l 4- m 2
m , — em: | m x + em2
+ m 2
*
[ + m 2 J
Where mi = mass of body 1
• m2 = mass of body 2
• vi = velocity of body 1
• v2 = velocity of body 2= velocity of body 1 after collision= velocity of body 2 
after collision
Where e = coefficient restitution
• In case of head-on elastic collision e = 1
• In case of head-on inelastic collision 0 < e < 1
• In case of head-on perfectly inelastic collision e = 0
The procedure for analyzing a collision depends on whether the process 
is elastic or inelastic. Kinetic energy is conserved in elastic collisions, whereas 
kinetic energy is converted into other forms of energy during an inelastic collision. 
In both types of collisions, momentum is conserved.
• Elastic Collisions
° Some kinetic energy is converted into sound energy when pool balls 
collide otherwise, the collision would be silent and a very small amount 
of kinetic energy is lost to friction.
° However, the dissipated energy is such a small fraction of the ball’s 
kinetic energy that we can treat the collision as elastic.
Equations for Kinetic Energy and Linear Momentum
• Let's examine an elastic collision between two particles of mass
, respectively. Assume that the collision is head-on, so we are dealing with 
only one dimension— you are unlikely to find two-dimensional collisions of any 
complexity on SAT II Physics. The velocities of the particles before the elastic 
collision are
l’i
and
V ,
, respectively. The velocities of the particles after the elastic collision are
V
and
'Y
. Applying the law of conservation of kinetic energy, we find:
Applying the law of conservation of linear momentum:
m it» i + — miiV +
• These two equations put together will help you solve any problem involving 
elastic collisions. Usually, you will be given quantities for
mi
and
, and can then manipulate the two equations to solve for
V,’
and
»Y
• A head-on with the cue ball in pool, Both of these balls have the same mass, 
and the velocity of the cue ball is initially
. What are the velocities of the two balls after the collision? Assume the 
collision is perfectly elastic
L'sto re
a ft©
Substituting
m | = m j = m
and
v 2 = 0
into the equation for conservation of kinetic energy we find:
-mpj = -nj(t)!/2 + I'i?)
V ? = v f + Vi?
Applying the same substitutions to the equation for conservation of momentum, we 
find:
rnt'i = m v\l + m vjl 
V t = vt'+i V
If we square this second equation, we get:
t’| = vi? + V 2 > l + 2 i'|/r> 2 /
By subtracting the equation for kinetic energy from this equation, we get:
2i.'i/rV = 0
• The only way to account for this result is to conclude that
V\ - 0
and consequently.
vt = vi
Inelastic Collisions
• Most collisions are inelastic because kinetic energy is transferred to other 
forms of energy-such as thermal energy, potential energy, and sound-during 
the collision process.
• The kinetic energy is not conserved, in inelastic collision. Momentum is 
conserved in all inelastic collisions.
• The one exception to this rule is in the case of completely inelastic collisions.
• Completely Inelastic Collisions
° A completely inelastic collision also called a "perfectly" or "totally" 
inelastic collision, Is one in which the colliding objects stick together 
upon impact.
o As a result, the velocity of the two colliding objects is the same after 
they collide, 
o Because
l>,' = v 2’ = v '
, question may be asked for finding
for eg. below two gumballs, of mass m and mass 2m respectively, collide head-on. 
Before impact, the gumball of mass m is moving with a velocity
, and the gumball of mass 2m Is stationary.
b e fo re
• First, note that the gumball wad has a mass of m + 2m = 3m. The law of 
conservation of momentum tells us that
m v | = im v ‘
, and so
v ' = v t/3
. Therefore, the final gumball wad moves in the same direction as the first 
gumball, but with one-third of its velocity.
Virtual work
Virtual Displacement & Virtual Work
• If a set of particles or a body is in equilibrium under the action of forces then 
there is no motion and consequently, there is no actual displacement.
• Suppose that the set of particles or the body receives an imaginary 
displacement, the forces acting thereon being regarded as constant during 
the displacement. Then such a displacement is called virtual displacement 
and work done by the forces during such a displacement is called virtual 
work.
• It may be noted that a virtual displacement is only a hypothetical 
displacement involving no passage of time and is quite different from the 
actual displacement of a moving body taking place in the course of time.