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But because particle 2 is initially at rest, this equation becomes

m 1 v 1 x = m 1 v 1 x + m 2 v 2 x .

The components of the velocities along the x size 12{x} {} -axis have the form v cos θ size 12{v`"cos"`θ} {} . Because particle 1 initially moves along the x size 12{x} {} -axis, we find v 1 x = v 1 .

Conservation of momentum along the x size 12{x} {} -axis gives the following equation:

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

where θ 1 size 12{θ rSub { size 8{1} } } {} and θ 2 size 12{θ rSub { size 8{2} } } {} are as shown in [link] .

Conservation of momentum along the x size 12{x} {} -axis

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

Along the y size 12{y} {} -axis, the equation for conservation of momentum is

p 1 y + p 2 y = p 1 y + p 2 y


m 1 v 1 y + m 2 v 2 y = m 1 v 1 y + m 2 v 2 y .

But v 1 y is zero, because particle 1 initially moves along the x size 12{x} {} -axis. Because particle 2 is initially at rest, v 2 y is also zero. The equation for conservation of momentum along the y size 12{y} {} -axis becomes

0 = m 1 v 1 y + m 2 v 2 y .

The components of the velocities along the y size 12{y} {} -axis have the form v sin θ size 12{v`"sin"`θ} {} .

Thus, conservation of momentum along the y size 12{y} {} -axis gives the following equation:

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

Conservation of momentum along the y size 12{y} {} -axis

0 = m 1 v 1 sin θ 1 + m 2 v 2 sin θ 2

The equations of conservation of momentum along the x size 12{x} {} -axis and y size 12{y} {} -axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level.

Determining the final velocity of an unseen object from the scattering of another object

Suppose the following experiment is performed. A 0.250-kg object m 1 is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass of 0.400 kg m 2 size 12{ left (m rSub { size 8{2} } right )} {} . The 0.250-kg object emerges from the room at an angle of 45 . size 12{"45" "." 0°} {} with its incoming direction.

The speed of the 0.250-kg object is originally 2.00 m/s and is 1.50 m/s after the collision. Calculate the magnitude and direction of the velocity ( v 2 and θ 2 ) of the 0.400-kg object after the collision.


Momentum is conserved because the surface is frictionless. The coordinate system shown in [link] is one in which m 2 size 12{m rSub { size 8{2} } } {} is originally at rest and the initial velocity is parallel to the x size 12{x} {} -axis, so that conservation of momentum along the x size 12{x} {} - and y size 12{y} {} -axes is applicable.

Everything is known in these equations except v 2 and θ 2 , which are precisely the quantities we wish to find. We can find two unknowns because we have two independent equations: the equations describing the conservation of momentum in the x - and y -directions.


Solving 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 v 2 sin θ 2 and taking the ratio yields an equation (because tan θ = sin θ cos θ in which all but one quantity is known:

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

Entering known values into the previous equation gives

tan θ 2 = 1 . 50 m/s 0 . 7071 1 . 50 m/s 0 . 7071 2 . 00 m/s = 1 . 129 . size 12{"tan"θ rSub { size 8{2} } = { { left (1 "." "50"" m/s" right ) left (0 "." "7071" right )} over { left (1 "." "50"" m/s" right ) left (0 "." "7071" right ) - 2 "." "00" "m/s"} } = - 1 "." "129"} {}


θ 2 = tan 1 1 . 129 = 311 . 312º . size 12{θ rSub { size 8{2} } ="tan" rSup { size 8{ - 1} } left ( - 1 "." "129" right )="311" "." 5° approx "312"°} {}

Angles are defined as positive in the counter clockwise direction, so this angle indicates that m 2 is scattered to the right in [link] , as expected (this angle is in the fourth quadrant). Either equation for the x - or y -axis can now be used to solve for v 2 , but the latter equation is easiest because it has fewer terms.

Questions & Answers

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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