Collisions
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Elastic Collisions
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Elastic Collisions
1 dimensional collision
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Elastic Collisions
1 dimensional collision: bodies of equal mass
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Elastic Collisions
1 dimensional collision: bodies of equal mass
(one body initially stationary)
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Elastic Collisions
1 dimensional collision: bodies of equal mass
(one body initially stationary)
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A
B
uA
Before collision, the total momentum is equal to
the momentum of body A
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A
B
vB
After collision, the total momentum is equal to the
momentum of body B
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The principle of conservation of momentum states
that the total momentum after collision equal to the
total momentum before collision (assuming no
external forces acting on the bodies)
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The principle of conservation of momentum states
that the total momentum after collision equal to the
total momentum before collision (assuming no
external forces acting on the bodies)
mAuA = mBvB
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The principle of conservation of momentum states
that the total momentum after collision equal to the
total momentum before collision (assuming no
external forces acting on the bodies)
mAuA = mBvB
so, if the masses are equal the velocity of B after
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The principle of conservation of momentum states
that the total momentum after collision equal to the
total momentum before collision (assuming no
external forces acting on the bodies)
mAuA = mBvB
so, if the masses are equal the velocity of B after is
equal to the velocity of A before
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Bodies of different mass
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A
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B
A
B
uA
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Before the collision, the total momentum is equal
to the momentum of body A
A
B
uA
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A
vA
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B
vB
After the collision, the total momentum is the sum
of the momenta of body A and body B
A
vA
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B
vB
If we want to calculate the velocities, vA and vB we
will use the
A
vA
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B
vB
If we want to calculate the velocities, vA and vB we
will use the principle of conservation of momentum
A
vA
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B
vB
The principle of conservation of momentum can be
stated here as
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The principle of conservation of momentum can be
stated here as
mAuA = mAvA + mBvB
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The principle of conservation of momentum can be
stated here as
mAuA = mAvA + mBvB
If the collision is elastic then
© D Hoult 2010
The principle of conservation of momentum can be
stated here as
mAuA = mAvA + mBvB
If the collision is elastic then kinetic energy is also
conserved
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The principle of conservation of momentum can be
stated here as
mAuA = mAvA + mBvB
If the collision is elastic then kinetic energy is also
conserved
½ mAuA2 = ½ mAvA2 + ½ mBvB2
© D Hoult 2010
The principle of conservation of momentum can be
stated here as
mAuA = mAvA + mBvB
If the collision is elastic then kinetic energy is also
conserved
½ mAuA2 = ½ mAvA2 + ½ mBvB2
mAuA2 = mAvA2 + mBvB2
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mAuA = mAvA + mBvB
mAuA2 = mAvA2 + mBvB2
From these two equations, vA and vB can be found
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mAuA = mAvA + mBvB
mAuA2 = mAvA2 + mBvB2
From these two equations, vA and vB can be found
BUT
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It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
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It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
* a very useful phrase !
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It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
uA
In this case, the velocity of A relative to B, before
the collision is equal to
© D Hoult 2010
It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
uA
In this case, the velocity of A relative to B, before
the collision is equal to uA
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It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
vA
vB
and the velocity of B relative to A after the collision
is equal to
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It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
vA
vB
and the velocity of B relative to A after the collision
is equal to vB – vA
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It can be shown* that for an elastic collision, the
velocity of body A relative to body B before the
collision is equal to the velocity of body B relative
to body A after the collision
vA
vB
and the velocity of B relative to A after the collision
is equal to vB – vA
for proof click here
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We therefore have two easier equations to “play
with” to find the velocities of the bodies after the
collision
equation 1
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We therefore have two easier equations to “play
with” to find the velocities of the bodies after the
collision
equation 1
mAuA = mAvA + mBvB
equation 2
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We therefore have two easier equations to “play
with” to find the velocities of the bodies after the
collision
equation 1
mAuA = mAvA + mBvB
equation 2
uA = vB – vA
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A
B
uA
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A
B
uA
Before the collision, the total momentum is equal
to the momentum of body A
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A
vA
B
vB
After the collision, the total momentum is the sum
of the momenta of body A and body B
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Using the principle of conservation of momentum
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Using the principle of conservation of momentum
mAuA = mAvA + mBvB
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Using the principle of conservation of momentum
mAuA = mAvA + mBvB
A
vA
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B
vB
Using the principle of conservation of momentum
mAuA = mAvA + mBvB
A
vA
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B
vB
Using the principle of conservation of momentum
mAuA = mAvA + mBvB
A
vA
B
vB
One of the momenta after collision will be a
negative quantity
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2 dimensional collision
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2 dimensional collision
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A
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B
A
B
Before the collision, the total momentum is equal
to the momentum of body A
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After the collision, the total
momentum is equal to the sum
of the momenta of both bodies
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Now the sum must be a
vector sum
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mAvA
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mAvA
mBvB
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mAvA
mBvB
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mAvA
mBvB
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mAvA
mBvB
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mAvA
p
mBvB
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mAvA
p
mAuA
mBvB
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mAvA
p
mAuA
mBvB
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mAvA
p
mAuA
mBvB
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mAvA
p
mAuA
mBvB
p = mAuA
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2 dimensional collision: Example
Body A has initial speed uA = 50 ms-1
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2 dimensional collision: Example
Body A has initial speed uA = 50 ms-1
Body B is initially stationary
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2 dimensional collision: Example
Body A has initial speed uA = 50 ms-1
Body B is initially stationary
Mass of A = mass of B = 2 kg
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2 dimensional collision: Example
Body A has initial speed uA = 50 ms-1
Body B is initially stationary
Mass of A = mass of B = 2 kg
After the collision, body A is found to be moving at
speed vA = 25 ms-1 in a direction at 60° to its
original direction of motion
© D Hoult 2010
2 dimensional collision: Example
Body A has initial speed uA = 50 ms-1
Body B is initially stationary
Mass of A = mass of B = 2 kg
After the collision, body A is found to be moving at
speed vA = 25 ms-1 in a direction at 60° to its
original direction of motion
Find the kinetic energy possessed by body B after
the collision
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