# 8.4 Elastic collisions in one dimension  (Page 6/4)

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• Describe an elastic collision of two objects in one dimension.
• Define internal kinetic energy.
• Derive an expression for conservation of internal kinetic energy in a one dimensional collision.
• Determine the final velocities in an elastic collision given masses and initial velocities.

Let us consider various types of two-object collisions. These collisions are the easiest to analyze, and they illustrate many of the physical principles involved in collisions. The conservation of momentum principle is very useful here, and it can be used whenever the net external force on a system is zero.

We start with the elastic collision of two objects moving along the same line—a one-dimensional problem. An elastic collision    is one that also conserves internal kinetic energy. Internal kinetic energy is the sum of the kinetic energies of the objects in the system. [link] illustrates an elastic collision in which internal kinetic energy and momentum are conserved.

Truly elastic collisions can only be achieved with subatomic particles, such as electrons striking nuclei. Macroscopic collisions can be very nearly, but not quite, elastic—some kinetic energy is always converted into other forms of energy such as heat transfer due to friction and sound. One macroscopic collision that is nearly elastic is that of two steel blocks on ice. Another nearly elastic collision is that between two carts with spring bumpers on an air track. Icy surfaces and air tracks are nearly frictionless, more readily allowing nearly elastic collisions on them.

## Elastic collision

An elastic collision    is one that conserves internal kinetic energy.

## Internal kinetic energy

Internal kinetic energy is the sum of the kinetic energies of the objects in the system.

Now, to solve problems involving one-dimensional elastic collisions between two objects we can use the equations for conservation of momentum and conservation of internal kinetic energy. First, the equation for conservation of momentum for two objects in a one-dimensional collision is

${p}_{1}+{p}_{2}={p\prime }_{1}+{p\prime }_{2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left({F}_{\text{net}}=0\right)$

or

${m}_{1}{v}_{1}+{m}_{2}{v}_{2}={m}_{1}{v\prime }_{1}+{m}_{2}{v\prime }_{2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left({F}_{\text{net}}=0\right),$

where the primes (') indicate values after the collision. By definition, an elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals the sum after the collision. Thus,

$\frac{1}{2}{m}_{1}{{v}_{1}}^{2}+\frac{1}{2}{m}_{2}{{v}_{2}}^{2}=\frac{1}{2}{m}_{1}{{v\prime }_{1}}^{2}+\frac{1}{2}{m}_{2}{{v\prime }_{2}}^{2}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{(two-object elastic collision)}$

expresses the equation for conservation of internal kinetic energy in a one-dimensional collision.

## Calculating velocities following an elastic collision

Calculate the velocities of two objects following an elastic collision, given that

${m}_{1}=0\text{.}\text{500 kg,}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{m}_{2}=3\text{.}\text{50 kg,}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{v}_{1}=4\text{.}\text{00 m/s, and}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{v}_{2}=0\text{.}$

Strategy and Concept

First, visualize what the initial conditions mean—a small object strikes a larger object that is initially at rest. This situation is slightly simpler than the situation shown in [link] where both objects are initially moving. We are asked to find two unknowns (the final velocities ${v\prime }_{1}$ and ${v\prime }_{2}$ ). To find two unknowns, we must use two independent equations. Because this collision is elastic, we can use the above two equations. Both can be simplified by the fact that object 2 is initially at rest, and thus ${v}_{2}=0$ . Once we simplify these equations, we combine them algebraically to solve for the unknowns.

hi guys can you teach me how to solve a logarithm?
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
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I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
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yes of course Villa flor
David
equations of kinematics for constant acceleration
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well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
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the latter is true
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plz hu can explain Heisenberg's uncertainty principle
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ok
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how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
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i think the acceleration is 0 since the car does not change its speed unless there are other conditions
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yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
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who can help me with a relative density question
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1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
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morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
no
Emmanuel
hi what is physical education?
Kate
BPED..is my course.
Kate
No
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I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
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how to calculate velocity
v=d/t
Emeka
Villaflor
Villaflor
v=d/t
Nisha
hello bro hw is life with you
Mine is good. How about you?
Chase
Hi room of engineers
yes,hi sir
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hello
akinmeji
Hello
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hello
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hi
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hi
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so, what is going on here
akinmeji
Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
wow, you got this w/o knowing math
Thomas
I guess that's it
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later people
Thomas
mathematics is everywhere
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thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
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A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
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16.66 meters?
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16.71m2
aways
,i have a question of let me give answer
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the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
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difference between charge and force
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define oscillation
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upward force exerted by water on a body.
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A hydrometer of mass 0.15kg and of uniform cross sectional area 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
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solve for height
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wenhe
stages of matter.
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States of matter are solid liquid gas and plasma
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what are Newtown's law
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An object as rest will remain at rest or in motion will remain at motion. Force is mass times acceleration. And a force will have equal and opposite reaction.
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do we have any thing like plasma as a states of matter
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no
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Plasma is a state of matter. It's much later on. The sun for example is not a gas ball. Its plasma.
wenhe
why is plasma a state of matter
Lila
In lower grades you are taught that there are 3 states of matter. But later on, you'll learn that if you eat gas enough. It turns to plasma
wenhe
Heat*
wenhe
yes we have plasma as a state of matter
akinmeji
it is anything that occupie space and has mass
Anand
I never head of that before
daniel
Anything that occupies space and has mass is matter.
wenhe
what is the difference between vapor and gas
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Plasma is best described as an ionized gas because it is made up of positively and negatively charged particles.
akinmeji
Vapour is gas.
wenhe
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akinmeji
was bose Einstein discovered by Albert Einstein
Lila
We have like 6 states or more. But those are newly discovered.
wenhe
OK, thanks at list I have experience about that one now.
daniel
Einstein​ predicted it
wenhe
With another person
wenhe
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Lila
Fermionic condensate
wenhe
OK what is Albert Einstein
daniel
And maybe more. I just rmb those. Tbh for school you only need to know 4.
wenhe
Albert Einstein​is the name of a scientist
wenhe
OK
daniel
what is the physical state of water and salt
Lila
can I ask off topic question?
Kristine
Both are matter. So can theoretically exist in all those states. But in room temperature, water is liquid and salt is solid.
wenhe
Yes
wenhe
Kristine
I'm not very good at maths. But if I can answer... Lol
wenhe
The reflector of a radiotelescope is in the shape of a parabola revolve its axis, if the diameter of the reflector is 400ft. above the vertex of the parabola, what should be the depth of reflector?
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Are there any other factors given?
wenhe
Or just that?
wenhe
matter is merely energy condensed to a slow vibration
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Cuz with those factors, I can't solve it.
wenhe
none
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Well. There must be another factor given or something. Like an equation of the parabola. Or else I can't solve it
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explain why water and salt are compounds
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That's chem. But both water and salt are made of more than 1 element. So they are compounds.
wenhe
Compound meaning 2 or more things added together.
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deficiency of vitamin E cause ?
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I don't know if this is the equation or formula your looking for; (x-h)²=4c(y-k) (x-h)²=-4c(y-k) (y-k)²=-4c(x-h) (y-k)²=4c(x-h)
Kristine
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Kristine