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What else can we learn by examining the equation x = x 0 + v 0 t + 1 2 at 2 ? size 12{x=x rSub { size 8{0} } +v rSub { size 8{0} } t+ { {1} over {2} } ital "at" rSup { size 8{2} } } {} We see that:

  • displacement depends on the square of the elapsed time when acceleration is not zero. In [link] , the dragster covers only one fourth of the total distance in the first half of the elapsed time
  • if acceleration is zero, then the initial velocity equals average velocity ( v 0 = v - size 12{v rSub { size 8{0} } = { bar {v}}} {} ) and x = x 0 + v 0 t + 1 2 at 2 size 12{x=x rSub { size 8{0} } +v rSub { size 8{0} } t+ { {1} over {2} } ital "at" rSup { size 8{2} } } {} becomes x = x 0 + v 0 t size 12{x=x rSub { size 8{0} } +v rSub { size 8{0} } t} {}

Solving for final velocity when velocity is not constant ( a 0 )

A fourth useful equation can be obtained from another algebraic manipulation of previous equations.

If we solve v = v 0 + at size 12{v=v rSub { size 8{0} } + ital "at"} {} for t size 12{t} {} , we get

t = v v 0 a . size 12{t= { {v - v rSub { size 8{0} } } over {a} } "." } {}

Substituting this and v - = v 0 + v 2 size 12{ { bar {v}}= { {v rSub { size 8{0} } +v} over {2} } } {} into x = x 0 + v - t size 12{x=x rSub { size 8{0} } + { bar {v}}t} {} , we get

v 2 = v 0 2 + 2 a x x 0 ( constant a ) . size 12{v rSup { size 8{2} } =v rSub { size 8{0} } rSup { size 8{2} } +2a left (x - x rSub { size 8{0} } right )" " \( "constant "a \) "." } {}

Calculating final velocity: dragsters

Calculate the final velocity of the dragster in [link] without using information about time.

Strategy

Draw a sketch.

Acceleration vector arrow pointing toward the right, labeled twenty-six point zero meters per second squared. Initial velocity equals 0. Final velocity equals question mark.

The equation v 2 = v 0 2 + 2 a ( x x 0 ) is ideally suited to this task because it relates velocities, acceleration, and displacement, and no time information is required.

Solution

1. Identify the known values. We know that v 0 = 0 size 12{v rSub { size 8{0} } =0} {} , since the dragster starts from rest. Then we note that x x 0 = 402 m size 12{x - x rSub { size 8{0} } ="402 m"} {} (this was the answer in [link] ). Finally, the average acceleration was given to be a = 26 . 0 m/s 2 size 12{a="26" "." "0 m/s" rSup { size 8{2} } } {} .

2. Plug the knowns into the equation v 2 = v 0 2 + 2 a ( x x 0 ) and solve for v .

v 2 = 0 + 2 26 . 0 m/s 2 402 m . size 12{v rSup { size 8{2} } =0+2 left ("26" "." "0 m/s" rSup { size 8{2} } right ) left ("402 m" right )} {}

Thus

v 2 = 2 . 09 × 10 4 m 2 /s 2 . size 12{v rSup { size 8{2} } =2 "." "09" times "10" rSup { size 8{4} } `m rSup { size 8{2} } "/s" rSup { size 8{2} } } {}

To get v size 12{v} {} , we take the square root:

v = 2 . 09 × 10 4 m 2 /s 2 = 145 m/s .

Discussion

145 m/s is about 522 km/h or about 324 mi/h, but even this breakneck speed is short of the record for the quarter mile. Also, note that a square root has two values; we took the positive value to indicate a velocity in the same direction as the acceleration.

Got questions? Get instant answers now!

An examination of the equation v 2 = v 0 2 + 2 a ( x x 0 ) size 12{v rSup { size 8{2} } =v rSub { size 8{0} } rSup { size 8{2} } +2a \( x - x rSub { size 8{0} } \) } {} can produce further insights into the general relationships among physical quantities:

  • The final velocity depends on how large the acceleration is and the distance over which it acts
  • For a fixed deceleration, a car that is going twice as fast doesn’t simply stop in twice the distance—it takes much further to stop. (This is why we have reduced speed zones near schools.)

Putting equations together

In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed.

Summary of kinematic equations (constant a size 12{a} {} )

x = x 0 + v - t size 12{x=`x rSub { size 8{0} } `+` { bar {v}}t} {}
v - = v 0 + v 2 size 12{ { bar {v}}=` { {v rSub { size 8{0} } +v} over {2} } } {}
v = v 0 + at size 12{v=v rSub { size 8{0} } + ital "at"} {}
x = x 0 + v 0 t + 1 2 at 2 size 12{x=x rSub { size 8{0} } +v rSub { size 8{0} } t+ { {1} over {2} } ital "at" rSup { size 8{2} } } {}
v 2 = v 0 2 + 2 a x x 0 size 12{v rSup { size 8{2} } =v rSub { size 8{0} } rSup { size 8{2} } +2a left (x - x rSub { size 8{0} } right )} {}

Calculating displacement: how far does a car go when coming to a halt?

On dry concrete, a car can decelerate at a rate of 7 . 00 m/s 2 size 12{7 "." "00 m/s" rSup { size 8{2} } } {} , whereas on wet concrete it can decelerate at only 5 . 00 m/s 2 size 12{5 "." "00 m/s" rSup { size 8{2} } } {} . Find the distances necessary to stop a car moving at 30.0 m/s (about 110 km/h) (a) on dry concrete and (b) on wet concrete. (c) Repeat both calculations, finding the displacement from the point where the driver sees a traffic light turn red, taking into account his reaction time of 0.500 s to get his foot on the brake.

Strategy

Draw a sketch.

Initial velocity equals thirty meters per second. Final velocity equals 0. Acceleration dry equals negative 7 point zero zero meters per second squared. Acceleration wet equals negative 5 point zero zero meters per second squared.

In order to determine which equations are best to use, we need to list all of the known values and identify exactly what we need to solve for. We shall do this explicitly in the next several examples, using tables to set them off.

Questions & Answers

hello friends what is hadronic heating systems
Rabilu Reply
Hydronics is the use of a liquid heat-transfer medium in heating and cooling systems. 
Balogun
what is mass
Victor Reply
is the amount of an object
Sendawula
mass is the measure of the inertia of a body
Ishmeal
advantages of CRO over ordinary voltmeter
Dismas Reply
what is the difference between displacement and distance?!
Daniel Reply
what is equilibrium
Sade Reply
If a system is said to be under equilibrium whenever there is no force act upon it... And it remain in its initial stage..
soniya
what is velocity
Ahmed
time rate of displacement of a body is called velocity
muhammad
velocity is the gradient of acceleration time graph
Etana
actually equilibrium is when a body is in total balance where in no external force is acting on it. Or the forces on the left hand side equal those on the left hand side and downward forces equal upward forces & anticlockwise moment equal clockwise moment about the same point.
Etana
I mean left hand side and right hand side
Etana
What is conductivity
Saud Reply
It is the ease with which electrical charges or heat can be transmitted through a material or a solution.
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how to find magnitude and direction
Arjune Reply
how to caclculate for speed
Arjune
derivation of ohms law
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derivation of resistance
Kazeem
R=v/I where R=resistor, v=voltage, I=current
Kazeem
magnitude
Arjune
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
Electric current is the flow of electrons
Kelly Reply
is there really flow of electrons exist?
babar
Yes It exists
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babar
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babar
an electron will flow accross a conductor because or when it posseses kinectic energy
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electron can not flow jist trasmit electrical energy
ghulam
free electrons of conductor
ankita
electric means the flow heat current.
Serah Reply
electric means the flow of heat current in a circuit.
Serah
What is electric
Manasseh Reply
electric means?
ghulam
electric means the flow of heat current in a circuit.
Serah
electric means the flow of electric current through conductor
Sade
the continuos flow of electrons in a circuit is called electric
ANUBHA
electric means charge
ghulam
electric means current
Sade
flow of current.
Sendawula
a boy cycles continuously through a distance of 1.0km in 5minutes. calculate his average speed in ms-1(meter per second). how do I solve this
Jenny Reply
speed = distance/time be sure to convert the km to m and minutes to seconds check my utube video "mathwithmrv speed"
PhysicswithMrV
d=1.0km÷1000=0.001 t=5×60=300s s=d\t s=0.001/300=0.0000033m\s
Serah
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
D=1km=1000m t=5mins×60secs=300sec s=d/t=3.333m/s
Daniel
I think Daniel Glorious is ryt
Amalia
why we cannot use DC instead of AC in a transformer
kusshaf Reply
becuse the d .c cannot travel for long distance trnsmission
ghulam
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Chiwetalu Reply
branch of science which deals with matter energy and their relationship between them
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Life science
the
what is heat and temperature
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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