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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

or

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.

Strategy

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.

Solution

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"} {}

Thus,

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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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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