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Collisions
Dr. Loai Afana
247
Collisions
In general, a “collision” is an interaction in which
•
two objects strike one another
•
the net external impulse is zero or negligibly small (momentum is conserved)
Examples:

Two billiard balls colliding on a billiards table

An alpha particle colliding with a heavy atom

Two galaxies colliding
Dr. Loai Afana
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Types of Collisions
1- Collisions in One Dimension
2- Tow Dimensional Collisions
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Dr. Loai Afana
1- Collisions in One Dimension
[1] Elastic collision:
Both momentum and kinetic energy are conserved
– Perfectly elastic collisions occur on a microscopic level
– In macroscopic collisions, only approximately elastic collisions actually occur
We can apply:
conservation of momentum and
conservation of kinetic energy:

P  Cons tan t
 Pi   Pf

m1v1i  m2v2i  m1v1 f  m2v2 f
K  1 mv 2
2

Dr. Loai Afana
250
[2] inelastic collision:
-kinetic energy is not conserved although momentum is still conserved
-If the objects stick together after the collision, it is a perfectly inelastic collision
We can apply:
conservation of momentum.

P  Cons tan t
 Pi   Pf

m1v1i  m2v2i  m1v1 f  m2v2 f
Dr. Loai Afana
251
Perfectly Inelastic Collisions
•
Since the objects stick together, they share the same velocity after the collision
We can apply:
conservation of momentum.

P  Cons tan t
P  P
i
f
m1v1i  m2 v2i  m1v1 f  m2v2 f
: v1 f  v2 f  v f
m1v1i  m2v2i  (m1  m2 )v f



In an inelastic collision, some kinetic energy is lost, but the objects do not stick together
Elastic and perfectly inelastic collisions are limiting cases, most actual collisions fall in between these two types
Dr. Loai Afana
252
2- Tow Dimensional Collisions
For an isolated system, the total momentum is conserved. For a general two-dimensional
collision of two particles, this implies that the total momentum in each direction is conserved.

P  Cons tan t
P
i
( p1i )x  ( p2i )x

P
f
 ( p1 f
)  (p )
x
2f
x
( p1i ) y  ( p2i ) y  ( p1 f )y  ( p2 f )y
As an example, let us consider a two dimensional problem in which a particle of mass m1
collides with a particle of mass m2, initially at rest. After the collision, m1 moves at an angle
θ with respect to the x-axis and m2 moves at an angle φ with respect to the x-axis .
Discussion
Dr. Loai Afana
254
Example: A ball with a mass of 1.2 kg moving to the right at 2.0 m/s collides with a ball of
mass 1.8 kg moving at 1.5 m/s to the left. If the collision is an elastic collision,
what are the velocities of the balls after the collision?
Remark:
The speed v can be positive or negative: v is positive if the body moves to the right,
and is negative if the body moves to the left.
Dr. Loai Afana
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Example:
A 3kg sphere makes a perfectly inelastic collision with a second sphere initially
at rest. The composite system moves with a speed equal to 1/3rd the original speed
of the 3 kg sphere.
What is the mass of the second sphere?
Dr. Loai Afana
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Example:
A 1500-kg car traveling east with speed of 25m/s collides at an intersection with a
2500-kg van traveling north with a speed of 20m/s as shown in Figure 9.8.
Find the final velocity after collision, if both cars moved together after collision.:
(perfectly inelastic collision )
.
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Dr. Loai Afana
259
Two blocks of masses M and 3M are placed on a horizontal, frictionless surface.
A light spring is attached to one of them, and the blocks are pushed together with
the spring between them (Fig.). A cord initially holding the blocks together is burned;
after this, the block of mass 3M moves to the right with a speed of 2.00 m/s.
(a) What is the speed of the block of mass M?
(b) Find the original potential energy in the spring if M = 0.350 kg.
Dr. Loai Afana
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0
A person with a mass of 50 kg slides down a hill of 0.46m high.
If he starts sliding from rest, and has a speed of 2.6 m/s at the bottom,
a) How much thermal energy is lost due to friction ? ( The work done by friction ).
b) If the hill has an angle of 20° above the horizontal what was the frictional force?
Dr. Loai Afana
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Example:
A 5kg block is set into motion up an inclined plane as in Figure with an
initial speed of 8 m/s. The block comes to rest after travelling 3 m along
the plane, as shown in the diagram. The plane is inclined at an angle of 30'
to the horizontal.
(a) Determine the change in kinetic energy.
(b) Determine the change in potential energy.
(c) Determine the frictional force on the block.
(d) What is the coefficient of kinetic friction?
2.5m
1m
Dr. Loai Afana
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Example:
The ambulance (mass 3000kg) shown in the Figure(2) slides (wheels locked) down a
frictionless incline that is 10 m long. It starts from rest at point A, and continues
along a rough surface until it comes to a complete stop at point C. If the coefficient
of kinetic friction between the ambulance and the horizontal rough surface is 0.1.
(a) Calculate the speed of the ambulance at point B.
(b) Compute the distance d the ambulance slides on the horizontal rough surface
before stopping.
Dr. Loai Afana
263
Example:
A block with a mass of 3 kg starts at a height h = 60cm on a plane with an inclination
angle of 30', as shown in Figure. At reaching the bottom of the slop, the block slides
along a horizontal surface. If the coefficient of friction on both surfaces is ,µk= 0.20,
how far will the block slide on the horizontal surface before coming to rest?
Dr. Loai Afana
264
Use conservation of energy to determine the final speed of a mass of 5.0kg
attached to a light cord over a mass less, frictionless pulley and attached to
another mass of 3.5 kg when the 5.0 kg mass has fallen (starting from rest) a
distance of 2.5 m as shown in Figure
2
6
Dr. Loai Afana
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Example:
a 1.7 kg block sliding with a speed of 1.7 m/s, the block collides with a spring
and compresses it 0.35 m before coming to rest.
What is the spring constant ?
Dr. Loai Afana
267
Example:
A 0.5 kg block is used to compresses a spring (with a spring constant of 80.0 N/m ) a distance
of 2.0 cm. After the spring is released, what is the final speed of the block?
Dr. Loai Afana
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Example:
A block having a mass of 0.80kg is given an initial velocity vA = 1.2m/s to the right
and collides with a spring of negligible mass and a spring constant k = 50N/m,
(A) Assuming the surface to be frictionless,
calculate the maximum compression of the spring after the collision.
(B) Suppose a constant force of kinetic friction acts between the block and the
surface, with μk = 0.50. If the speed of the block at the moment it collides
with the spring is vA = 1.2 m/s, what is the maximum compression xC in the
spring?
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Example:
In the pendulum experiment :
a bullet of mass 0.06 kg is fired horizontally into a wooden block of mass .2 kg.
The wooden block is suspended from the ceiling by a long string as shown in the diagram.
The collision is perfectly inelastic and after impact the bullet and the block swing together
until the block is 12 m above it's initial position.
Find:
a) the velocity of the bullet and block just after impact and
b) the velocity of the bullet just before impact
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Example:
As shown in Figure , a bullet of mass m and speed v passes completely
through a pendulum bob of mass M. The bullet emerges with a speed of V/2.
The pendulum bob is suspended by a stiff rod of length L and negligible mass.
What is the value of V such that the pendulum bob will swing through a
complete vertical circle only ?
f
Dr. Loai Afana
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Two blocks are free to slide along the frictionless wooden track ABC shown in
Figure. The block of mass m1 = 5.00 kg is released from A. and makes a
head on elastic collision with a block of mass m2 =10.0 kg, initially at rest at
point B. Calculate the maximum height to which m1 rises after the elastic
collision..
Dr. Loai Afana
272