re mainly three equations of motion which describe the relationship between velocity, time, acceleration and displacement.First, consider a body moving in a straight line with uniform acceleration. Then, let the initial velocity be u, acceleration be a, time period be t, velocity be v, and the distance travelled be S.The equation of motions derivation can be done in three ways which are:Derivation of equations of motion by Simple Algebraic MethodDerivation of Motion by Graphical MethodDerivation of Motion by CALCULUS MethodBelow, the equations of motion are derived by all the three methods in a simple and easy to understand way.Derivation of First Equation of MotionThe first equation of motion is:v = u + atDerivation of First Equation of Motion by Algebraic MethodIt is KNOWN that the acceleration (a) of the body is defined as the rate of change of velocity.So, the acceleration can be written as:a = v − utFrom this, rearranging the terms, the first equation of motion is obtained, which is:v = u + atDerivation of First Equation of Motion by Graphical MethodConsider the diagram of the velocity-time graph of a body below:Derivation Of Equation Of MotionIn this, the body is moving with an initial velocity of u at point A. The velocity of the body then changes from A to B in time t at a uniform rate. In the above diagram, BC is the final velocity i.e. v after the body travels from A to B at a uniform acceleration of a. In the graph, OC is the time t. Then, a perpendicular is drawn from B to OC, a parallel line is drawn from A to D, and another perpendicular is drawn from B to OE (represented by dotted lines).Following details are obtained from the graph above:The initial velocity of the body, u = OAThe final velocity of the body, v = BCFrom the graph,BC = BD + DCSo, v = BD + DCv = BD + OA (since DC = OA)Finally, v = BD + u (since OA = u) (Equation 1)Now, since the slope of a velocity-time graph is equal to acceleration a,So,a = slope of line ABa = BD/ADSince AD = AC = t, the above equation becomes:BD = at (Equation 2)Now, combining Equation 1 & 2, the following is obtained:v = at + uDerivation of First Equation of Motion by Calculus MethodIt is known that,Derivation Of Equation Of MotionSo,Derivation Of Equation Of MotionDerivation of Second Equation of MotionThe second equation of motion is:S = ut + ½ a2Derivation of Second Equation of Motion by Algebraic MethodConsider the same notations for the derivation of the second equation of motion by simple algebraic method.Derivation Of Second Equation Of MotionDerivation of Second Equation of Motion by Graphical MethodTaking the same diagram used in first law derivation:Derivation Of Equation Of MotionIn this diagram, the distance travelled (S) = Area of figure OABC = Area of rectangle OADC + Area of triangle ABD.Now, the area of the rectangle OADC = OA × OC = utAnd, Area of triangle ABD = (1/2) × Area of rectangle AEBD = (1/2) at2 (Since, AD = t and BD = at)Thus, the total distance covered will be:S = ut + (1/2) at2Derivation of Second Equation of Motion by Calculus MethodVelocity is the rate of change of displacement.Mathematically, this is expressed as\(v=\frac{ds}{dt}\)Rearranging the equation, we get\(ds=vdt\)Substituting the first equation of motion in the above equation, we get\(ds=(u+at)dt\) \(=(udt+at\,dt)\) \(\int_{0}^{s}ds=\int_{0}^{t}u\,dt+\int_{0}^{t}at\,dt\) \(s=ut+\frac{1}{2}at^2\)Derivation of Third Equation of MotionThe third equation of motion is:v2 = u2 + 2aSDerivation of Third Equation of Motion by Algebraic MethodDerivation Of Third Equation Of MotionDerivation of Third Equation of Motion by Graphical MethodDerivation Of Equation Of MotionThe total distance travelled, S = Area of trapezium OABC.So, S= 1/2(SumofParallelSides)×HeightS=(OA+CB)×OCSince, OA = u, CB = v, and OC = tThe above equation becomesS= 1/2(u+v)×tNow, since t = (v – u)/ aThe above equation can be written as:S= 1/2(u+v)×(v-u)/aRearranging the equation, we getS= 1/2(v+u)×(v-u)/aS = (v2-u2)/2aThird equation of motion is obtained by solving the above equation:v2 = u2+2aSDerivation of Third Equation of Motion by Calculus MethodIt is known that,Derivation Of Equation Of MotionThese were the detailed derivations for equations of motion in the graphical method, algebraic method and calculus method.Equations of Motion FormulaEquations of motion FormulaFirst equation of motion v=u+atSecond equation of motion \(s=ut+\frac{1}{2}at^{2}\)Third equation of motion v2 = u2+2as