Download 7th set - Nathan Dawson

Momentum
• Momentum is related to Newton’s first law that
states an object in motion will continue in
motion, an object at rest will remain at rest
unless an outside force acts upon it. (The more
momentum an object has, the greater the force
needed to stop it.)
• Momentum is related to Newton’s second law in
that F=ma=Dp/Dt.
• Not surprisingly, Newton’s Third law leads to the
law of CONSERVATION OF MOMENTUM
Definition of momentum
The linear momentum (or “momentum” for short) of an object is
defined as the product of its mass and its velocity.
MOMENTUM IS A VECTOR
The momentum is related to the force through Newton’s second law,
Consider these situations:
(i) a ball moving at speed v is brought to rest;
(ii) the same ball is projected from rest so it moves at speed v;
(iii) the same ball moving at speed v is bounced backward at
almost its original speed.
In which case(s) does the ball undergo the largest magnitude
change in momentum?
1. (i)
2. (i) and (ii)
3. (i), (ii), and (iii)
4. (iii)
Consider two carts, of masses m and 2m, at rest on an air
track. If you push first one cart for 3 s and then the other
for the same length of time, exerting equal force on each,
the momentum of the light cart is
1. four times
2. twice
3. equal to
4. one-half
5. one-quarter
the momentum of the heavy cart.
Impulse
From Newton’s second law, the net force on an object is equal to the rate of
change of its momentum
Multiply both sides of the equation by the time interval gives
We define the left hand side of the equation as
the impulse,
The average force acting
over a very brief time interval
gives the same impulse as
the actual force.
A ball hits a wall as shown in the figure. In which
direction is the impulse of the wall on the ball?
(Choose from A-G)
A tennis ball of mass m = 0.060 kg and speed v = 28.0 m/s
strikes a wall at a 45° angle and rebounds with the same
speed at 45°. What is the impulse (magnitude and
direction) given to the ball?
If the ball made contact with the wall for a period
of Dt = 20 ms, then what is the average force that
the wall exerts on the ball?
We are usually interested in the CHANGES in momentum:
A bug collides with the windshield of a car traveling
down the highway. Which experiences a greater force?
Which experiences a greater change in momentum?
1.
2.
3.
4.
5.
it depends on the initial velocity of the bug
the car
the bug
both experience the same change
none of the above
A soccer ball (m = 0.070 kg) approaches a player in a horizontal
line with a speed of 6.0 m/s. After it is kicked by the player the
ball is driven straight backward with a speed of 18 m/s. The ball
remains in contact with the foot for 0.60 s.
Calculate the impulse exerted on the ball by the player.
Calculate the average force exerted on the ball by the player
during the contact time.
Suppose a ping-pong ball and a bowling ball
are rolling toward you. Both have the same
momentum, and you exert the same force to
stop each. How do the time intervals to stop
them compare?
1. It takes less time to stop the ping-pong ball.
2. Both take the same time.
3. It takes more time to stop the ping-pong ball.
Conservation of Momentum
If no net external force acts on a system, the total momentum of the
system is a conserved quantity.
or
This can be rewritten in terms of the mass and the velocity of the objects colliding.
Conservation of Momentum & Newton’s 3rd law
During the collision, the objects exert forces on each other
There must be an equal and opposite force in the same
time interval,
where
Adding the two forces together gives
Therefore,
or
Momentum is conserved in an
ISOLATED system
Which of these systems are isolated?
A. While slipping on a patch of ice, a car collides
and sticks to another car. System: both cars
B. Same situation as in A. System: slipping car
C. A single car slips on a patch of ice. System: car
D. A car makes an emergency stop on a road.
System: car
E. A ball drops to Earth. System: ball
F. A billiard ball collides elastically with another
billiard ball on a pool table. System: both balls
A child in a boat throws a 5.30 kg package out horizontally
with a speed of 10.0 m/s. Calculate the velocity of the boat
immediately after, assuming it was initially at rest. The mass
of the child is 24.0 kg and the mass of the boat is 35.0 kg.
A car accelerates from rest. In doing so the
car gains a certain amount of momentum,
and in the opposite direction, the Earth gains
1. more momentum.
2. the same amount of momentum.
3. less momentum.
4. The answer depends on the amount of
friction between the two.
Suppose you are on a cart, initially at rest on a track with very
little friction. You throw balls at a partition that is rigidly
mounted on the cart. If
the balls bounce straight
back as shown in the
figure, is the cart put in
motion?
1. Yes, it moves to the right.
2. Yes, it moves to the left.
3. No, it remains in place.
4. It depends on the mass of the ball and the cart, and the
velocity of the throw.
Elastic & Inelastic collisions
In an elastic collision, the kinetic energy is also conserved during the collision.
and
In an inelastic collision, the kinetic energy is NOT conserved during the collision.
but
The kinetic energy that is lost is changed into other forms of energy,
often thermal energy, so that the total energy (as always) is conserved.
ELASTIC collisions in 1 dimension
(elastic collisions are the only time mechanical energy is conserved)
We can write down the conservation of momentum equation
We can write down the conservation of energy equation
Rearranging the conservation of energy equation and the conservation of
momentum equation via algebraic methods gives
and
Simplifying the conservation of energy equation and dividing by the conservation of
momentum equation gives
Thus, we arrive at a simplified equation for the elastic collision from the
conservation of energy,
Two billiard balls of equal mass undergo a perfectly elastic
head-on collision. If one ball’s initial speed was 2.00 m/s
and the other’s was 3.60 m/s in the opposite direction, what
will be their speeds and directions after the collision?
Collisions in two dimensions
Car One is traveling due north and Car Two is traveling due
east. After the collision shown, Car One rebounds due
south. Which of the numbered arrows is the only one that
can represent the final direction of Car Two?
A car with a mass of mC = 950 kg travels east and collides with a
truck with a mass of mT = 1900 kg that is traveling north that did
not stop at red light. The two vehicles stick together as a result of
the collision, and the wreckage slides at vf = 16.0 m/s in the
direction q = 24.0o east of north.
Determine the initial speed of the car, vCi.
Determine the initial speed of the truck, vTi.
Determine the change in kinetic energy
before and after the collision, DK.
An object with mass m1 = 1.00 kg has a velocity of
= 2.00 m/s î
+ 2.00 m/s ĵ. It collides with a second object of mass m2 = 2.00kg that
is initially at rest. After the collision we measure the first block to
have a magnitude of velocity
= 1.50 m/s at an angle of q1 = 330o
as measured counter-clockwise from the x-axis.
Before
collision
(a) What is the magnitude of velocity,
After
collision
, of the second block?
(b) What is the angle, q2, after the collision that the trajectory of the second block
makes as measured counter-clockwise from the x-axis?
A cart moving at speed v collides with an identical
stationary cart on an airtrack, and the two stick
together after the collision. What is their velocity
after colliding?
1. v
2. 0.5 v
3. zero
4. –0.5 v
5. –v
6. need more information
A 50.0-kg boy runs at a speed of 10.0 m/s and jumps
onto a cart as shown. The cart is initially at rest. If the
speed of the cart with the boy on it is 2.50 m/s, what is
the mass of the cart?
(1) 150 kg (2) 200 kg (3) 300 kg
(4) 175 kg (5) 100 kg
A block of mass m is fired with a velocity vbi. The bullet
collides and sticks inside a block of mass M initially at
rest and hanging from the ceiling by a massless string of
known length.
Find the maximum angle, q, that the pendulum makes
with respect to its equilibrium position.
before
after
Explosions are just inelastic collisions run in
reverse.
Momentum must be conserved if all parts of
the exploding object are included in the
system.
An object initially at rest with a mass of mtot = 6.5 kg explodes
into two fragments, the first has a mass of mA = 4.0 kg, and the
other has a mass of mB = 2.5 kg. The explosion perfectly transfers
the energy Q = 150 J into kinetic energy of the fragments.
Calculate the kinetic energy of the object with mass mA after the
explosion.
Calculate the kinetic energy of the object with mass mB after the
explosion.
Center of mass/momentum
N
xcm

v cm
m1x1 + m2 x2 + m3 x3 + m4 x 4 + ...
=
=
m1 + m2 + m3 + m4 + ...




m1v1 + m2 v 2 + m3 v 3 + m4 v 4 + ...
=
=
m1 + m2 + m3 + m4 + ...
mi xi

i=1
N
mi

i=1
N

i=1

mi vi
N
mi

i=1
=

p
M
On a long lightweight (air-filled) raft, three people of roughly equal
mass m sit along the x axis at positions xA = 1.0 m, xB = 5.0 m, and
xC = 6.0 m measured from the left-hand end as shown in the figure.
Find the position of the center-of-mass.
Applying Newton’s Second Law to an
assembly of objects (that may or may
not be connected!)
(not connected)
(connected)
Torque
Rotating an object requires a force applied.
The magnitude of the torque is the applied force in the perpendicular direction to the
distance from the axis of rotation, multiplied by the distance from the axis of rotation.
From the geometry in the above diagrams, we can see that
. Thus,
and
A semi-truck with tyres having a radius of 0.500 m is
stopped at a stop light. When the light turns green, the
driver steps on the accelerator which applies a force
between the bottom of a tyre and the pavement of 2600 N
in the horizontal direction.
Determine the magnitude of the torque on the tyre from
the force applied by the pavement.
Calculate the net torque about the axle of the wheel
shown in the below figure. Assume that a friction
torque of 0.60 mN opposes the motion.