Momentum
p = m v
SI unit for momentum: kg m s-1 (no special name). Show that kg m
s-1 = Ns
Momentum is a vector quantity Velocity and momentum vectors
point in the same direction. Calculate the momentum of (i) a Bus: m
= 9000 kg; v = 16 m /s
(ii) Car: m = 1800 kg; v = 80 m /s
(iii) Train: m = 3.6 104 kg; v = 4 m /s
The train, bus, and car all have different masses and speeds,
but their momenta are the same in magnitude. The massive train has
a slow speed; the low-mass car has a great speed; and the bus has
moderate mass and speed. Note: We can only say that the magnitudes
of their momenta are equal Something big and slow could have the
same momentum as something small and fast. A car possesses 20 000
units of momentum. What would be the car's new momentum if ...a.
its velocity was doubled.b. its velocity was tripled.c. its mass
was doubled (by adding more passengers and a greater load)d. both
its velocity was doubled and its mass was doubled.
Force and momentumAs the force acts upon the object for a given
amount of time, the object's velocity is changed; and hence, the
object's momentum is changed. Any object with momentum is going to
be hard to stop. To stop such an object, it is necessary to apply a
force against its motion for a given period of time. The more
momentum that an object has, the harder that it is to stop. Thus,
it would require a greater amount of force or a longer amount of
time or both to bring such an object to a halt.
A net force is required to change a bodys momentum.F=ma
F =
An unbalanced force always accelerates an object - either
speeding it up or slowing it down. If the force acts opposite the
object's motion, it slows the object down. If a force acts in the
same direction as the object's motion, then the force speeds the
object up. Either way, a force will change the velocity of an
object.Application - Car safety featuresWhen there is a car crash,
the car, its contents and the passengers decelerate rapidly. They
experience great forces because of the change in momentum, which
can cause injury.Modern cars also have safety features that absorb
kinetic energy in collisions. These typically include: Seat belts
Air bags Crumple zonesThese features reduce injuries to the people
in the car by absorbing energy when they change shape. They
increase the time taken for the change in momentum on the
occupants' bodies, reducing the forces involved and injuries.
Seat beltsSeat belts stop you tumbling around inside the car if
there is a collision. However, they are designed to stretch a bit
in a collision. This increases the time taken for the body's
momentum to reach zero, so reduces the forces on it.Air bagsAir
bags increase the time taken for the head's momentum to reach zero,
so reduce the forces on it. They also act a soft cushion and
prevent cuts.Crumple zonesCrumple zones are areas of a vehicle that
are designed to crush in a controlled way in a collision. They
increase the time taken to change the momentum in a crash, which
reduces the force involved.Questions1. A dart of mass 200g strikes
a dartboard with a horizontal velocity of 6m/s. The dart pushes its
way into the dart board a little and takes 0,05s to stop moving.
Calculate the magnitude of the average force that the dartboard
exerts on the dart to stop it.
2. A trolley A of mass 3kg travels due east at a velocity of
7m.s-1. Another trolley B of mass 4kg follows along the same path
at 12m.s-1 and collides with it. After the collision, trolley A
continues to travel east but with a velocity of 11m.s-1. Calculate
the velocity of trolley B after the collision.
3. A car of mass 1100kg moves at 24 m/s. What is the braking
force needed to bring the car to a halt in 2.0 seconds?
Conservation of momentumThe principle of conservation of linear
momentum states that in any interaction between bodies, linear
momentum is conserved provided on resultant external force acts on
the bodies.MOMENTUM - in sticky collisions:
Question1. 30 kg child running at 7 m/s jumps onto a 10 kg sled
which was initially at rest. What will be the velocity of the
child+sled immediately after the child jumps on the sled?
2. A 70 kg astronaut floating in an orbiting space station
throws a 1.0 kg water bottle across the room at a speed of 8.0 m/s.
What is the magnitude of the astronauts recoil velocity? Assuming
that this is a perfect inelastica. collision, calculate the
velocity after theb. collision in the example below.
3.
4. A 1000 kg car is rolling down the street at 2.5 m/s. How fast
would a 2500 kg car have to collide into it in order to bring it to
rest (0 m/s)?
5. A 3000-kg truck moving rightward with a speed of 5 km/hr
collides head-on with a 1000-kg car moving leftward with a speed of
10 km/hr. The two vehicles stick together and move with the same
velocity after the collision. Determine the post-collision speed of
the car and truck.
6. 45 cm/s when it encounters a smaller fish (m=0.25 kg) that is
at rest. The large fish swallows the smaller fish and continues in
motion at a reduced speed. Find its new speed
Impulse F = m a
or
F = m v / tIf both sides of the above equation are multiplied by
the quantity t, a new equation results.
F t = m vImpulse = Change in momentum
The last statement of Newtons second law, together with the
third law, tells us what happens in a situation where two ice
skaters push each other apart as they move horizontally across the
ice.
The push of the man on the woman is equal and opposite to the
push of the woman on the man (Newtons 3rd law),
so the term Ft, which is called the impulse of the force, will
be exactly the same size during each interval t, as they push
apart.
As a consequence the overall change of momentum of the man will
be equal but opposite to the change of momentum of the woman. We
have proved that the conservation of momentum in this case is a
consequence of Newtons 2nd and 3rd laws.
Question: 1. Explain the instant where a tennis racket hits a
ball.
2. How much force is exerted on a 13 kg object accelerating from
20 m/s to 12 m/s in 3.5 seconds?
3. How much force is exerted on a 3 kg object accelerating from
12 m/s to rest in 5.5 seconds?
4. How much time is needed to accelerate a 3 kg object from rest
to 12 m/s if a force of 2.5-N is applied?
5. How much time is needed to accelerate a 4 kg object from 10
m/s to 12 m/s if a force of 2.5-N is applied?
6. An alpha particle is emitted fro a Po atom at a speed of 1.8
X 10-7 ms-1 the relative mass of alpha and the remaining nucleus
are 4.002 and 212.0, calculate the recoil velocity of the
nucleus.
7. It takes you all of 0.018 s to initially touch and then catch
a 0.600 kg football travelling at 16.0 m/s.(a) What is the change
in momentum for the football? (b) What is the impulse? (c) What is
the force that must be exerted to stop the ball?
8. A tennis ball is at rest when it experiences a forward force
to set it in motion. It then strikes a wall where it encounters a
force that slows it down and finally turns it around and sends it
backwards.
9.
10. A hockey player hits a hockey ball of mass 150 g moving
directly towards her at 10m.s-1, straight back. The force which the
hockey stick exerts on the hockey ball is given by the graph
below.
Are of a force time graphSince the area under the line is really
just multiplying force and time, the area must be equal to the
impulse acting on the object.If we calculate the area under the
graph (a triangle) we will know what the impulse is.A = bh= (5.78
s)(3012 N)= 8704.68A = 8.70 x 103 kgm/sChange in momentum = 8.70 x
103 kgm/s
Questions1. Calvin & Hobbes, 45 kg, are sleigh riding down a
hill. The hill they are on is irregular shaped, slopes up and down
and has snow of varying depths and textures. Below is a force vs
time graph of the force acting on their sled. The initial velocity
when they hit the part of the hill depicted on the graph at 11
m/s.
(a) How do you find the impulse from 30 to 60 seconds?(b) Which
30 seconds time interval contains a net negative impulse?(c) What
is the impulse from 30 to 60 seconds?(d) What is the impulse from
90 to 120 seconds?(e) What is the impulse from 150 to 180
seconds?(f) Using the information from text above combined with the
graph, calculate the final velocity at the 60 second mark.(g)
Calculate the final velocity at the 120 second mark.(h) Calculate
the final velocity at the end of the ride?(i) What was the average
velocity for the entire ride?(j) What is the (average) acceleration
over the entire ride?(k) What must the initial velocity be so that
Calvin and Hobbes come to a rest at the end of the ride?
2. To the right is a force versus time graph for a childs toy
car. The toy is malfunctioning and is producing the force
shown,
a. What is velocity of the toy car, 0.756 kg, after 20 seconds
if it starts from rest?b. What is velocity of the toy car, 0.756
kg, after 20 seconds if it starts from 10 m/s?
3. A 1.00 kg ball traveling towards a soccer player at a
velocity of 5.00 m/s rebounds off the soccer players foot at a
velocity of 8.50 m/s. If the time of contact between the ball and
the players foot was 2.00 x 10-2 seconds, what was the force that
the foot applied on the ball?
4. A 1.50 kg rock falls from the top of a 10.0 m high building
and strikes the ground below. What is the force of the ground
acting on the rock if it comes to a stop in 0.350 seconds.
5. A girl of mass 50kg wearing skates stands stationary on ice.
She throws a ball of mass 400g at a speed of 25m/s. Calculate: a.
her velocity immediately after throwing the ball. (4) b. her
momentum immediately after throwing the ball. (3) c. the momentum
of the ball immediately after it has been thrown. (3)6. In order to
complete a tough split in bowling, a bowling ball of mass 6.5 kg is
rolled with a velocity of 12 m/s. The ball hits a 1.3 kg bowling
pin, sending it off with a speed of 3.5 m/s at an angle of 80
degrees with respect to the original direction of the bowling ball.
What is the angle and direction of the bowling balls velocity after
the collision with the pin?
RocketsThe equation F = can be use if the motion of
rockets.Where the stream of matter (waste gas) is ejected at a
speed of v as mass decreases it rocket gains speed.Question1. A
rocket is ready to take off. It ejects waste gases, the result of
exploding a mixture of oxygen and hydrogen. At a rate of 1.2 x 104
kgs-1. If the exhaust speed is 4.0 x 103 ms-1 calculate the upward
trust of the ejected gases on the rocket.
Momentum and energySometimes it's desirable to express the
kinetic energy of a particle in terms of the momentum. That's easy
enough. Since v = and the kinetic energy so
Note that if a massive particle and a light particle have the
same momentum, the light one will have a lot more kinetic energy.
If a light particle and a heavy one have the same velocity, the
heavy one has more kinetic energy. Calculate the momentum of a 58g
tennis ball with KE 75jCalculate the KE of an arrow of mass 0.12kg
at a momentum of 5.2kgms-1.What is the momentum of an electron of
mass 9.1 x 10-31kg and a proton of mass 1.7 x 10-27kg, it each of
which has a KE of 150eV.Elastic ans in eelstic collitions
A perfectly elastic collision is defined as one in which there
is no loss of kinetic energy in the collision. An inelastic
collision is one in which part of the kinetic energy is changed to
some other form of energy in the collision. Any macroscopic
collision between objects will convert some of the kinetic energy
into internal energy and other forms of energy, so no large scale
impacts are perfectly elastic. Momentum is conserved in inelastic
collisions, but one cannot track the kinetic energy through the
collision since some of it is converted to other forms of energy.
Collisions in ideal gases approach perfectly elastic collisions, as
do scattering interactions of sub-atomic particles which are
deflected by the electromagnetic force.
pa + pb = pa + pb pa + pb = pa + pbKEa + KEb = KEa + KEb KEa +
KEb = KEa + KEb + Heat
1. Two V2FV1FV1IV2Itrain cars of identical mass (m = 10.0 kg)
collide and move off as shown. The left car was initially traveling
at 2.0 m/s right and the right car was initially traveling at a
velocity of 3.0 m/s left. After the collision, the left car is
traveling at a velocity of 2.5 m/s to the left.
a. What type of a collision is depicted here?
______________________________________________
b. What was the initial momentum of the right car?
c. What was the initial momentum of the left car?
d. What was the final momentum of the left car?
e. What was the final momentum of the right car?
A 1250 kg car is moving down the highway with a velocity of 32.0
m/s when it bumps into the car ahead of it which has a mass of 875
kg and a velocity of 25.0 m/s. After the collision, the two cars
stick together. What will be the resulting velocity of the two cars
together? How much energy will be lost in this collision? (
A 4.0 kg model rocket is launched, shooting 50.0 g of burned
fuel from its exhaust at an average velocity of 625 m/s. What is
the velocity of the rocket after the fuel has burned?
A 45.0 kg ice skater stands at rest on the ice. A friend tosses
the skater a 5.0 kg ball. The skater and the ball then move
backwards across the ice with a speed of 0.5 m/s. What was the
speed of the ball at the moment just before the skater caught it?
State whether the above equation is elastic or inelastic.Collisions
in two dimensionsThrough resolving
By assuming that the above is an elastic collision,
According to the conservation of momentum
X axis
Yaxis
By using the above 3 eqations the the unknown can be
calculated.
A proton of mass M collides obliquely with another proton. The
first proton is moving with a speed of 6.0 x 106 m/s before it hits
the second, stationary proton. Assuming the collision is perfectly
elastic, and using the fact that the first proton is moved 30 from
its initial path after the collision, figure out the speed and
direction of each proton after the collision. Note that after the
collision the two protons travel at 90 to each other.
A 0.40 kg model airplane is traveling 20 km/h toward the south.
A 0.50 kg model airplane traveling 25 km/h in a direction 20 east
of south collides with the first model airplane. The two planes
stick together on impact. What is the direction and magnitude of
the velocity of the combined wreckage immediately after the
collision?A steel ball of mass 10 kg moves due East at 5.0 m/s. It
collides with a rubber ball of mass 5.0 kg moving at 10 m/s due
North. After the collision the steel ball moves at an angle of 60
East of North with a speed of 4.0 m/s. What is the velocity of the
rubber ball after the collision?Two balls of equal mass m undergo a
collision. One ball is initially stationary. After the collision,
the velocities of the balls make angles of 31.1 and 48.9 relative
to the original direction of motion of the moving ball.a) If the
initial speed of the moving ball is 2.25 m/s, what are the speeds
of the balls after collision? b) Is this collision elastic?
Explain how an astronaut who is stranded in free space a short
distance from his spacecraft might employ his knowledge of momentum
to return safely to the craft. Why must he be very careful about
his momentum?
