Transcript Collisions: Impulse and Momentum

Collisions:
Impulse and
Momentum
AP Physics 1
M. Dimler
What to Expect from this Unit
1.
2.
3.
4.
5.
Momentum
Impulse
Impulse-Momentum Theorem
Conservation of Momentum
Collisions (Elastic, Inelastic, Perfectly Inelastic, and
Explosion)
6. Elastic Collisions-Conservation of KE
7. Collisions in 2-D
8. Center of Mass
Momentum
• A moving object’s momentum is the product of its
mass and velocity.
• Momentum is a vector quantity, and the direction
of momentum is in the direction of motion.
• Units of momentum – kg∙m/s
Momentum
• The rate of change in momentum is equal to the net
force.
• This can be shown using Newton’s 2nd Law.
Impulse
• Impulse is the change in an object’s momentum,
which is also equal to the net force on the object
multiplied by the time during which the force acts.
• Impulse is in the direction of net force.
• Impulse can be expressed in N∙s or kg∙m/s.
Impulse-Momentum Theorem
• Impulse can be calculated using the ImpulseMomentum Theorem
• Impulse can also be determined by quantifying the
area under a net force vs. time graph.
Impulse
• The impulse tells us that we
can get the same change in
momentum with a large
force acting for a short time,
or a small force acting for a
longer time.
• This is why you should bend
your knees when you land;
why airbags work; and why
landing on a pillow hurts less
than landing on concrete.
Practice Problem
• Force of a tennis serve: For a top
player, a tennis ball may leave the
racket on the serve with a speed of 55
m/s (about 120 mph). If the ball has a
mass of 0.060 kg and is in contact with
the racket for about 4 ms, estimate the
average force on the ball. Would this
force be large enough to lift a 60 kg
person?
Practice Problem
• A 2-kg coconut falls from the top of a
tall tree, 30m above a person’s head.
The coconut strikes and comes to rest
on the person’s head.
a) Calculate the magnitude of the momentum of the
coconut just before it hits the person in the head.
Practice Problem
• A 2-kg coconut falls from the top of a
tall tree, 30m above a person’s head.
The coconut strikes and comes to rest
on the person’s head.
b) Calculate the magnitude and direction of the impulse
experienced by the coconut in colliding with the person’s
head?
Practice Problem
• A 2-kg coconut falls from the top of a tall
tree, 30m above a person’s head. The
coconut strikes and comes to rest on the
person’s head.
c) The person’s head experienced a force of 10,000 N in the
collision. How long was the coconut in contact with the
person's head?
A) Much more than 10 seconds
B) Just a bit more than one second
C) Just a bit less than one second
D) Much less than 1/10 of a second
Practice Problem
• A 2-kg coconut falls from the top of a
tall tree, 30m above a person’s head.
The coconut strikes and comes to rest
on the person’s head.
d) In a different situation, explain how it could be possible for
an identical coconut dropped from the same height to
hit the person's head, but produce less than 10,000 N of
force.
Challenge Practice Problem
Bend your knees when landing:
a) Calculate the impulse experienced when a
70kg person lands on firm ground after jumping
from a height of 3.0m.
b) Estimate the average force exerted on the
person’s feet by the ground if the landing is stifflegged, and again…
c) … with bent legs.
Note: With stiff legs, assume the body moves 1.0cm during
impact, and when the legs are bent, about 50cm.
Conservation of Momentum
Note: In any system in which the only forces acting are between objects in that
system, momentum is conserved. This effectively means that momentum is
conserved in all collision.
Conservation of Momentum
• Law of Conservation of Momentum: The total momentum of an
isolated system of objects remains constant.
Conservation of Energy and
Momentum in Collisions
• Momentum is
conserved in all
collisions.
• Collisions in which
kinetic energy is
conserved as well
are called elastic
collisions, and those
in which it is not are
called inelastic
collisions.
Elastic Collisions
• Here we have two objects
colliding elastically. We
know the masses and the
initial speeds.
• Since both momentum and
kinetic energy are
conserved, we can write
two equations. This allows
us to solve for the two
unknown final speeds.
Inelastic Collisions
• With inelastic collisions,
some of the initial kinetic
energy is lost to thermal or
potential energy. It may
also be gained during
explosions, as there is the
addition of chemical or
nuclear energy.
• A completely inelastic
collision is one where the
objects stick together
afterwards, so there is only
one final velocity.
Explosion
(ex. Balloon, Rocket)
Momentum conservation works for a rocket as long as
we consider the rocket and its fuel to be one system,
and account for the mass loss of the rocket.
Practice Problem
A car on a freeway collides with a
mosquito, which was initially at rest.
Justify all answers thoroughly.
a) Did the total momentum of the car-mosquito system
increase, decrease, or remain the same after the collision?
b) Did the momentum of the mosquito increase, decrease,
or remain the same after the collision?
c) Did the momentum of the car increase, decrease, or
remain the same after the collision?
Practice Problem
A car on a freeway collides with a
mosquito, which was initially at rest. Justify
all answers thoroughly.
d) Which changed its speed by more in the collision, the car or
the mosquito? (Or did they change speed by the same
amount?)
e) Which changed its momentum by more in the collision, the
car or the mosquito? (Or did they change momentum by the
same amount?)
f) Which experienced a greater impulse in the collision, the car
or the mosquito? (Or did they experience the same impulse?)
Practice Problem
A car on a freeway collides with a
mosquito, which was initially at rest.
Justify all answers thoroughly.
g) Which experienced a greater magnitude of net force
during the collision, the car or the mosquito? (Or did they
experience the same net force?)
Practice Problem
• Ballistic Pendulum: The ballistic
pendulum is a device used to
measure the speed of a
projectile, such as a bullet. The
projectile, of mass m, is fired into
a large block of mass M, which is
suspended like a pendulum,
where M>m. As a result of the
collision, the pendulum and
projectile together swing up to a
max height, h. Determine the
relationship between the initial
horizontal speed of the projectile
(v) and the max height (h).
Group Quiz Problem
Car A has a mass of 1,500 kg and travels to
the right with a speed of 20 m/s. Car B initially
travels to the left with a speed of 10 m/s. After
the vehicles collide, they stick together,
moving left with a common speed of 5m/s.
Justify all answers thoroughly.
a) Calculate the mass of Car B.
b) This collision is not elastic. Explain why not.
c) Describe specifically a collision between these
two cars with the same initial conditions but
which is not elastic, and in which the cars
bounce off one another.
d) Is the collision elastic when Car B remains at rest
after the collision?
Collisions in 2-Dimensions (or 3)
• http://www.khanacademy.org/science/physics/line
ar-momentum/momentum-tutorial/v/2-dimensionalmomentum-problem
Center of Mass
• Momentum is a powerful concept not only for
analyzing collisions, but also for analyzing the
translational motion of real extended objects.
In (a), the diver’s motion is
pure translation; in (b) it is
translation plus rotation.
There is one point that moves
in the same path a particle
would take if subjected to
the same force as the diver.
This point is called the center
of mass (CM).
Center of Mass
“CM”
The general motion of an object can be
considered as the sum of the translational
motion of the CM, plus rotational, vibrational, or
other forms of motion about the CM.
Center of Mass
• For two particles, the center of mass lies
closer to the one with the most mass:
• where M is the total mass.
Center of Mass vs. Center of Gravity
• The center of gravity is
the point where the
gravitational force can
be considered to act.
It is the same as the
center of mass as long
as the gravitational
force does not vary
among different parts
of the object.
Experimentally Determine
Center of Gravity
• The center of gravity can be found experimentally by
suspending an object from different points. The CM need not
be within the actual object – a doughnut’s CM is in the center
of the hole.
Center of Mass and
Translational Motion
• The total momentum of a system of particles is
equal to the product of the total mass and the
velocity of the center of mass.
• Applying Newton’s 2nd Law: The sum of all the forces
acting on a system is equal to the total mass of the
system multiplied by the acceleration of the center
of mass
Center of Mass and
Translational Motion
This is particularly useful in the analysis of separations
and explosions; the center of mass (which may not
correspond to the position of any particle) continues
to move according to the net force.
Motion of the Center of Mass
• Example: Imagine that an astronaut on a
spacewalk throws a rope around a small asteroid,
and then pulls the asteroid toward him. Where will
the asteroid and the astronaut collide?
• Example: A toy rocket is in projectile motion, so
that it is on track to land 30m from its launch point.
While in the air, the rocket explodes into two
identical pieces, one of which land 35 m from the
launch point. Where does the first piece land?
Summary
• Momentum of an object:
• Newton’s second law:
• Total momentum of an isolated system of objects is
conserved.
• During a collision, the colliding objects can be
considered to be an isolated system even if external
forces exist, as long as they are not too large.
• Momentum will therefore be conserved during
collisions.
Summary
•
• In an elastic collision, total kinetic energy is also
conserved.
• In an inelastic collision, some kinetic energy is lost.
• In a completely inelastic collision, the two objects
stick together after the collision.
• The center of mass of a system is the point at which
external forces can be considered to act.
Bullet Block Experiment
Veritasium
• https://www.youtube.com/watch?v=vWVZ6APXM4w
• https://www.youtube.com/watch?v=BLYoyLcdGPc