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Section summary

  • The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the x -axis parallel to the velocity of the incoming particle.
  • Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the x -axis), stated by m 1 v 1 = m 1 v 1 cos θ 1 + m 2 v 2 cos θ 2 and along the direction perpendicular to the initial direction (the y -axis) stated by 0 = m 1 v 1 y + m 2 v 2 y .
  • The internal kinetic before and after the collision of two objects that have equal masses is
    1 2 mv 1 2 = 1 2 mv 1 2 + 1 2 mv 2 2 + mv 1 v 2 cos θ 1 θ 2 .
  • Point masses are structureless particles that cannot spin.

Conceptual questions

[link] shows a cube at rest and a small object heading toward it. (a) Describe the directions (angle θ 1 size 12{θ rSub { size 8{1} } } {} ) at which the small object can emerge after colliding elastically with the cube. How does θ 1 size 12{θ rSub { size 8{1} } } {} depend on b size 12{b} {} , the so-called impact parameter? Ignore any effects that might be due to rotation after the collision, and assume that the cube is much more massive than the small object. (b) Answer the same questions if the small object instead collides with a massive sphere.

A ball m one moves horizontally to the right with speed v one. It will collide with a stationary square labeled capital m two that is rotated at approximately forty-five degrees. The point of impact is on a face of the square a distance b above the center of the square. After the collision the ball is shown heading off at an angle theta one above the horizontal with a speed v one prime. The square remains essentially stationary (v 2 prime is approximately zero).
A small object approaches a collision with a much more massive cube, after which its velocity has the direction θ 1 size 12{θ rSub { size 8{1} } } {} . The angles at which the small object can be scattered are determined by the shape of the object it strikes and the impact parameter b size 12{b} {} .


Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of 30 . ,what is the velocity (magnitude and direction) of the second puck? (You may use the result that θ 1 θ 2 = 90º for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic.

(a) 3.00 m/s, 60º below x size 12{x} {} -axis

(b) Find speed of first puck after collision: 0 = m v 1 sin 30º m v 2 sin 60º v 1 = v 2 sin 60º sin 30º = 5.196 m/s

Verify that ratio of initial to final KE equals one: KE = 1 2 mv 1 2 = 18 m J KE = 1 2 mv 1 2 + 1 2 mv 2 2 = 18 m J KE KE′ = 1

Confirm that the results of the example [link] do conserve momentum in both the x size 12{x} {} - and y size 12{y} {} -directions.

A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of 20 . size 12{"20" "." 0°} {} above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired?

(a) 2 . 26 m/s size 12{ - 2 "." "26"`"m/s"} {}

(b) 7 . 63 × 10 3 J size 12{7 "." "63" times "10" rSup { size 8{3} } `J} {}

(c) The ground will exert a normal force to oppose recoil of the cannon in the vertical direction. The momentum in the vertical direction is transferred to the earth. The energy is transferred into the ground, making a dent where the cannon is. After long barrages, cannon have erratic aim because the ground is full of divots.

Professional Application

A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of 85 . size 12{"85" "." 0°} {} to the initial direction of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final velocity (magnitude and direction) of the bowling ball. (b) Is the collision elastic? (c) Linear kinetic energy is greater after the collision. Discuss how spin on the ball might be converted to linear kinetic energy in the collision.

Professional Application

Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei ( 4 He ) from gold-197 nuclei ( 197 Au ) . The energy of the incoming helium nucleus was 8.00 × 10 13 J , and the masses of the helium and gold nuclei were 6.68 × 10 27 kg and 3.29 × 10 25 kg , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of 120º during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?

(a) 1 . 50 × 10 7 m/s; 5 . 36 × 10 5 m/s at 29.5º

(b) 7 . 52 × 10 13 J size 12{7 "." "52" times "10" rSup { size 8{ - "13"} } `J} {}

Professional Application

Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1200 kg and is approaching at 8 . 00 m/s size 12{8 "." "00"`"m/s"} {} due south. The second car has a mass of 850 kg and is approaching at 17 . 0 m/s size 12{"17" "." 0`"m/s"} {} due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) Note that because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the x size 12{x} {} -axis and y size 12{y} {} -axis; instead, you must look for other simplifying aspects.

Starting with equations m 1 v 1 = m 1 v 1 cos θ 1 + m 2 v 2 cos θ 2 and 0 = m 1 v 1 sin θ 1 + m 2 v 2 sin θ 2 for conservation of momentum in the x - and y -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses,

1 2 mv 1 2 = 1 2 mv 1 2 + 1 2 mv 2 2 + mv 1 v 2 cos θ 1 θ 2

as discussed in the text.

We are given that m 1 = m 2 m size 12{m rSub { size 8{1} } =m rSub { size 8{2} } equiv m} {} . The given equations then become:

v 1 = v 1 cos θ 1 + v 2 cos θ 2


0 = v 1 sin θ 1 + v 2 sin θ 2 .

Square each equation to get

v 1 2 = v 1 2 cos 2 θ 1 + v 2 2 cos 2 θ 2 + 2 v 1 v 2 cos θ 1 cos θ 2 0 = v 1 2 sin 2 θ 1 + v 2 2 sin 2 θ 2 + 2 v 1 v 2 sin θ 1 sin θ 2 .

Add these two equations and simplify:

v 1 2 = v 1 2 + v 2 2 + 2 v 1 v 2 cos θ 1 cos θ 2 + sin θ 1 sin θ 2 = v 1 2 + v 2 2 + 2 v 1 v 2 1 2 cos θ 1 θ 2 + 1 2 cos θ 1 + θ 2 + 1 2 cos θ 1 θ 2 1 2 cos θ 1 + θ 2 = v 1 2 + v 2 2 + 2 v 1 v 2 cos θ 1 θ 2 .

Multiply the entire equation by 1 2 m size 12{ { { size 8{1} } over { size 8{2} } } m} {} to recover the kinetic energy:

1 2 mv 1 2 = 1 2 m v 1 2 + 1 2 m v 2 2 + m v 1 v 2 cos θ 1 θ 2

Integrated Concepts

A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
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