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  • Calculate displacement of an object that is not accelerating, given initial position and velocity.
  • Calculate final velocity of an accelerating object, given initial velocity, acceleration, and time.
  • Calculate displacement and final position of an accelerating object, given initial position, initial velocity, time, and acceleration.
Four men racing up a river in their kayaks.
Kinematic equations can help us describe and predict the motion of moving objects such as these kayaks racing in Newbury, England. (credit: Barry Skeates, Flickr)

We might know that the greater the acceleration of, say, a car moving away from a stop sign, the greater the displacement in a given time. But we have not developed a specific equation that relates acceleration and displacement. In this section, we develop some convenient equations for kinematic relationships, starting from the definitions of displacement, velocity, and acceleration already covered.

Notation: t , x , v , a

First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a stopwatch, is a great simplification. Since elapsed time is Δ t = t f t 0 , taking t 0 = 0 means that Δ t = t f , the final time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of position and velocity. That is, x 0 is the initial position and v 0 is the initial velocity . We put no subscripts on the final values. That is, t is the final time , x is the final position , and v is the final velocity . This gives a simpler expression for elapsed time—now, Δ t = t . It also simplifies the expression for displacement, which is now Δ x = x x 0 . Also, it simplifies the expression for change in velocity, which is now Δ v = v v 0 . To summarize, using the simplified notation, with the initial time taken to be zero,

Δ t = t Δ x = x x 0 Δ v = v v 0

where the subscript 0 denotes an initial value and the absence of a subscript denotes a final value in whatever motion is under consideration.

We now make the important assumption that acceleration is constant . This assumption allows us to avoid using calculus to find instantaneous acceleration. Since acceleration is constant, the average and instantaneous accelerations are equal. That is,

a - = a = constant , size 12{ { bar {a}}=a="constant"} {}

so we use the symbol a size 12{a} {} for acceleration at all times. Assuming acceleration to be constant does not seriously limit the situations we can study nor degrade the accuracy of our treatment. For one thing, acceleration is constant in a great number of situations. Furthermore, in many other situations we can accurately describe motion by assuming a constant acceleration equal to the average acceleration for that motion. Finally, in motions where acceleration changes drastically, such as a car accelerating to top speed and then braking to a stop, the motion can be considered in separate parts, each of which has its own constant acceleration.

Solving for displacement ( Δ x ) and final position ( x size 12{x} {} ) from average velocity when acceleration ( a size 12{a} {} ) is constant

To get our first two new equations, we start with the definition of average velocity:

v - = Δ x Δ t . size 12{ { bar {v}}= { {Δx} over {Δt} } "." } {}

Substituting the simplified notation for Δ x and Δ t yields

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

Solving for x size 12{x} {} yields

x = x 0 + v - t , size 12{x=x rSub { size 8{0} } + { bar {v}}t" " \( "constant a" \) ,} {}

where the average velocity is

v - = v 0 + v 2 ( constant a ) . size 12{ { bar {v}}= { {v rSub { size 8{0} } +v} over {2} } " " \( "constant "a \) "." } {}

Questions & Answers

how can we find absolute uncertainty
ayesha Reply
it what?
Luke
in physics
ayesha
the basic formula is uncertainty in momentum multiplied buy uncertainty In position is greater than or equal to 4×pi/2. same formula for energy and time
Luke
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ayesha
what
uma
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Suthar
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farooq
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Iram
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Ojeh Reply
what is the dimension of strain
Joy Reply
Is there a formula for time of free fall given that the body has initial velocity? In other words, formula for time that takes a downward-shot projectile to hit the ground. Thanks!
Cyclone Reply
hi
Agboro
hiii
Chandan
Hi
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hi
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Hy
Kulsum
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Kulsum
watt is the unit of power
Rahul
p=f.v
Rahul
watt can also be expressed as Nm/s
Rahul
what s i unit of mass
Maxamed
SI unit of mass is Kg(kilogram).
Robel
what is formula of distance
Maxamed
Formula for for the falling body with initial velocity is:v^2=v(initial)^2+2*g*h
Mateo
i can't understand
Maxamed
we can't do this calculation without knowing the height of the initial position of the particle
Chathu
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Imoreh
2 forces whose resultant is 100N, are at right angle to each other .if one of them makes an angle of 30 degree with the resultant determine it's magnitude
Victor Reply
50 N... (50 *1.732)N
Sahim
Plz cheak the ans and give reply..
Sahim
50 N...(50 *1.732)N
Ibrahim
Is earth is an inertial frame?
Sahim Reply
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
thanks
Irungu
Most welcome
Sahim
Hey.. I've a question.
Sahim Reply
Is earth inertia frame?
Sahim
only the center
Shii
What is an abucus?
Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash Reply
someone please give answer to this.
prakash
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
Ebenezer
please
Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
Oluwatola Reply
what is cesium atoms?
prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
I want learn physics
Vinodhini Reply
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Vinodhini
try to imagine everything you study in 3d
revolutionary
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Vinodhini
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Vinodhini
vernier caliper usage practically
Vinodhini
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Vinodhini
what are the solution to all the exercise..?
What is realm
Vinodhini Reply
The quantum realm, also called the quantum scale, is a term of art inphysics referring to scales where quantum mechanical effects become important when studied as an isolated system. Typically, this means distances of 100 nanometers (10−9meters) or less or at very low temperature.
revolutionary
How to understand physics
Vinodhini Reply
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Prince
by reading it
Austin
understanding difficult
Vinodhini
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Vinodhini
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revolutionary
what is mean by Newtonian principle of Relativity? definition and explanation with example
revolutionary Reply

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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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