collisions © d hoult 2010. elastic collisions © d hoult 2010
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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)
© D Hoult 2010
Elastic Collisions
1 dimensional collision: bodies of equal mass
(one body initially stationary)
© D Hoult 2010
© D Hoult 2010
Before collision, the total momentum is equal to the momentum of body A
A B
uA
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After collision, the total momentum is equal to the momentum of body B
A B
vB
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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
© D Hoult 2010
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 B
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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 B
vA vB
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A B
vA vB
After the collision, the total momentum is the sum of the momenta of body A and body B
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A B
vA vB
If we want to calculate the velocities, vA and vB we will use the
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A B
vA vB
If we want to calculate the velocities, vA and vB we will use the principle of conservation of momentum
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The principle of conservation of momentum can be stated here as
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mAuA = mAvA + mBvB
The principle of conservation of momentum can be stated here as
© D Hoult 2010
mAuA = mAvA + mBvB
If the collision is elastic then
The principle of conservation of momentum can be stated here as
© D Hoult 2010
mAuA = mAvA + mBvB
If the collision is elastic then kinetic energy is also conserved
The principle of conservation of momentum can be stated here as
© D Hoult 2010
mAuA = mAvA + mBvB
If the collision is elastic then kinetic energy is also conserved
½ mAuA2 = ½ mAvA
2 + ½ mBvB
2
The principle of conservation of momentum can be stated here as
© D Hoult 2010
mAuA = mAvA + mBvB
If the collision is elastic then kinetic energy is also conserved
mAuA2 = mAvA
2 + mBvB
2
½ mAuA2 = ½ mAvA
2 + ½ mBvB
2
The principle of conservation of momentum can be stated here as
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From these two equations, vA and vB can be found
mAuA = mAvA + mBvB
mAuA2 = mAvA
2 + mBvB
2
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From these two equations, vA and vB can be found
mAuA = mAvA + mBvB
mAuA2 = mAvA
2 + mBvB
2
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 !© 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
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
In this case, the velocity of A relative to B, before the collision is equal to uA
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
and the velocity of B relative to A after the collision is equal to
vA vB
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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
and the velocity of B relative to A after the collision is equal to vB – vA
vA vB
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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
for proof click here
and the velocity of B relative to A after the collision is equal to vB – vA
vA vB
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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
equation 2
mAuA = mAvA + mBvB
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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
equation 2
mAuA = mAvA + mBvB
uA = vB – vA
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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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After the collision, the total momentum is the sum of the momenta of body A and body B
A B
vA vB
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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 B
vA vB
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Using the principle of conservation of momentum
mAuA = mAvA + mBvB
A B
vA vB
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Using the principle of conservation of momentum
mAuA = mAvA + mBvB
A B
vA 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 B
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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© D Hoult 2010
Now the sum must be a vector sum
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mAvA
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mBvB
mAvA
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mBvB
mAvA
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mBvB
mAvA
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mBvB
mAvA
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p
mBvB
mAvA
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mBvB
mAvA
p
mAuA
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mBvB
mAvA
p
mAuA
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mBvB
mAvA
p
mAuA
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p = mAuA
mBvB
mAvA
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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