2.5 Motion equations for constant acceleration in one dimension  (Page 3/8)

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In addition to being useful in problem solving, the equation $v={v}_{0}+\text{at}$ gives us insight into the relationships among velocity, acceleration, and time. From it we can see, for example, that

• final velocity depends on how large the acceleration is and how long it lasts
• if the acceleration is zero, then the final velocity equals the initial velocity $\left(v={v}_{0}\right)$ , as expected (i.e., velocity is constant)
• if $a$ is negative, then the final velocity is less than the initial velocity

(All of these observations fit our intuition, and it is always useful to examine basic equations in light of our intuition and experiences to check that they do indeed describe nature accurately.)

Making connections: real-world connection

An intercontinental ballistic missile (ICBM) has a larger average acceleration than the Space Shuttle and achieves a greater velocity in the first minute or two of flight (actual ICBM burn times are classified—short-burn-time missiles are more difficult for an enemy to destroy). But the Space Shuttle obtains a greater final velocity, so that it can orbit the earth rather than come directly back down as an ICBM does. The Space Shuttle does this by accelerating for a longer time.

Solving for final position when velocity is not constant ( $a\ne 0$ )

We can combine the equations above to find a third equation that allows us to calculate the final position of an object experiencing constant acceleration. We start with

$v={v}_{0}+\text{at}.$

Adding ${v}_{0}$ to each side of this equation and dividing by 2 gives

$\frac{{v}_{0}+v}{2}={v}_{0}+\frac{1}{2}\text{at}\text{.}$

Since $\frac{{v}_{0}+v}{2}=\stackrel{-}{v}$ for constant acceleration, then

$\stackrel{-}{v}={v}_{0}+\frac{1}{2}\text{at}\text{.}$

Now we substitute this expression for $\stackrel{-}{v}$ into the equation for displacement, $x={x}_{0}+\stackrel{-}{v}t$ , yielding

$x={x}_{0}+{v}_{0}t+\frac{1}{2}{\text{at}}^{2}\phantom{\rule{0.25em}{0ex}}\left(\text{constant}\phantom{\rule{0.25em}{0ex}}a\right)\text{.}$

Calculating displacement of an accelerating object: dragsters

Dragsters can achieve average accelerations of $\text{26}\text{.}{\text{0 m/s}}^{2}$ . Suppose such a dragster accelerates from rest at this rate for 5.56 s. How far does it travel in this time?

Strategy

Draw a sketch.

We are asked to find displacement, which is $x$ if we take ${x}_{0}$ to be zero. (Think about it like the starting line of a race. It can be anywhere, but we call it 0 and measure all other positions relative to it.) We can use the equation $x={x}_{0}+{v}_{0}t+\frac{1}{2}{\text{at}}^{2}$ once we identify ${v}_{0}$ , $a$ , and $t$ from the statement of the problem.

Solution

1. Identify the knowns. Starting from rest means that ${v}_{0}=0$ , $a$ is given as $\text{26}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ and $t$ is given as 5.56 s.

2. Plug the known values into the equation to solve for the unknown $x$ :

$x={x}_{0}+{v}_{0}t+\frac{1}{2}{\text{at}}^{2}\text{.}$

Since the initial position and velocity are both zero, this simplifies to

$x=\frac{1}{2}{\text{at}}^{2}\text{.}$

Substituting the identified values of $a$ and $t$ gives

$x=\frac{1}{2}\left(\text{26}\text{.}{\text{0 m/s}}^{2}\right){\left(5\text{.}\text{56 s}\right)}^{2},$

yielding

$x=\text{402 m.}$

Discussion

If we convert 402 m to miles, we find that the distance covered is very close to one quarter of a mile, the standard distance for drag racing. So the answer is reasonable. This is an impressive displacement in only 5.56 s, but top-notch dragsters can do a quarter mile in even less time than this.

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
Afzal
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
Beatriz
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
Sobirjon
the latter is true
Sobirjon
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plz hu can explain Heisenberg's uncertainty principle
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ok
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i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
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
Luis
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
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hi what is physical education?
Kate
BPED..is my course.
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No
Emmanuel
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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.
Richard
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
Okwethu
hello
akinmeji
Hello
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hello
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hi
Sakhi
hi
H.C
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
Thomas
later people
Thomas
mathematics is everywhere
Anand
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
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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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oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door
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difference between charge and force
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akinmeji
define oscillation
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upward force exerted by water on a body.
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archimedes principle
Nicholas
mg=p(area x height)g
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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
wenhe
what are Newtown's law
Lila
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.
wenhe
do we have any thing like plasma as a states of matter
daniel
no
Nicholas
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
Lila
Plasma is best described as an ionized gas because it is made up of positively and negatively charged particles.
akinmeji
Vapour is gas.
wenhe
Aside plasma, we also have Bose- Einstein. this is also another state of matter
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?
Kristine
Are there any other factors given?
wenhe
Or just that?
wenhe
matter is merely energy condensed to a slow vibration
Vashist
Cuz with those factors, I can't solve it.
wenhe
none
Kristine
Well. There must be another factor given or something. Like an equation of the parabola. Or else I can't solve it
wenhe
explain why water and salt are compounds
Lila
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.
wenhe
salt chemically is combination of sodium n chloride whereas water is oxygen n hydrogen
Friday
deficiency of vitamin E cause ?
Friday
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Kristine
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Kristine