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Equations of Motion- Definition, Method, Formula

Aug 22, 2022

Deriving Equations of Motion



Equation of motion is mostly derived from the definition of the quantity which describes the motion and the time related to motion. 

The equations to be derived are: 

v = u + at 

s = ut + 1/2at2 

v2-u2 = 2as 


Deriving the equation in two types:  

  1. Algebraic method 
  1. Graphical method 

Deriving the equation v = u + at using the algebraic method: 

we know that acceleration is the ratio of change of velocity with time; hence, we have 

Deriving the equation v= u + at using the graphical method: 

Time 1

 Important points: 

  1. In the above graph, the y-axis represents velocity, and the x-axis represents time. 
  1. We can see initial velocity u is in the y-axis and also some points on v in the graph. 
  1. Therefore, the graph represents v-t graph, and the slope of the graph is acceleration.  
  1. The slope and the equation are derived as follows: 

Deriving the equation s = ut + 1/2 at2 using the algebraic method: 

we know that displacement = velocity x time  


From the first derivation, we know v = u + at 

Deriving the equation s = ut + 1/2 at2 using graphical method 

Time 2

Important motion: 

In the above graph, the total distance traveled in the graph is given by the area of RPQS 

The area under RPQS is further divided into the area of triangle RQS and the area of the rectangle RTOS. 

Hence, the distance traveled is, 

S = area of RQS + area RTOS 

S = ½ (RS x SQ) + RT x TO 

S=1/2 (t x at) + u x t 

S = ut +1/2 at2 

Deriving the equation v2 – u = 2as using the algebraic method: 

Deriving the equation v2 – u2 = 2as using the graphical method: 

Time 3

Important points: 

S =1/2(sum of opposite sides) x height 

S = (RS + PQ) x TO 

The above equation becomes 

S = 1/2(u + v) x (v – u)/a 

Rearranging the equation, we get 

S = 1/2(v + u) x (v – u) x a 

S = (v2 – u2)/2a 


v2 – u2 = 2as



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