Derivations of Equations of Motion (Graphically)

First Equation of Motion

Graphical Derivation of First Equation

Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t.

The figure shows the velocity-time graph of the motion of the object.

Slope of the v - t graph gives the acceleration of the moving object.

Thus, acceleration = slope = AB =

 

 

v - u = at

v = u + at  I equation of motion

Graphical Derivation of Second Equation

Distance travelled S = area of the trapezium ABDO

= area of rectangle ACDO + area of DABC

(v = u + at I eqn of motion; v - u = at)

Graphical Derivation of Third Equation

S = area of the trapezium OABD.

Substituting the value of t in equation (1) we get,

2aS = (v + u) (v - u)

(v + u)(v - u) = 2aS [using the identity a² - b² = (a+b) (a-b)]

v² - u² = 2aS  III Equation of Motion

Derivations of Equations of Motion (Graphically) First Equation of Motion Graphical Derivation of First Equation Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t. The figure shows the velocity-time graph of the motion of the object. Slope of the v - t graph gives the acceleration of the moving object. Thus, acceleration = slope = AB = v - u = at v = u + at I equation of motion Graphical Derivation of Second Equation Distance travelled S = area of the trapezium ABDO = area of rectangle ACDO + area of DABC (v = u + at I eqn of motion; v - u = at) Graphical Derivation of Third Equation S = area of the trapezium OABD. Substituting the value of t in equation (1) we get, 2aS = (v + u) (v - u) (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] v2 - u2 = 2aS III Equation of Motion

Derivations of Equations of Motion (Graphically) First Equation of Motion Graphical Derivation of First Equation Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t. The figure shows the velocity-time graph of the motion of the object. Slope of the v - t graph gives the acceleration of the moving object. Thus, acceleration = slope = AB = v - u = at v = u + at I equation of motion Graphical Derivation of Second Equation Distance travelled S = area of the trapezium ABDO = area of rectangle ACDO + area of DABC (v = u + at I eqn of motion; v - u = at) Graphical Derivation of Third Equation S = area of the trapezium OABD. Substituting the value of t in equation (1) we get, 2aS = (v + u) (v - u) (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] v2 - u2 = 2aS III Equation of Motion

Derivations of Equations of Motion (Graphically) First Equation of Motion Graphical Derivation of First Equation Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t. The figure shows the velocity-time graph of the motion of the object. Slope of the v - t graph gives the acceleration of the moving object. Thus, acceleration = slope = AB = v - u = at v = u + at I equation of motion Graphical Derivation of Second Equation Distance travelled S = area of the trapezium ABDO = area of rectangle ACDO + area of DABC (v = u + at I eqn of motion; v - u = at) Graphical Derivation of Third Equation S = area of the trapezium OABD. Substituting the value of t in equation (1) we get, 2aS = (v + u) (v - u) (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] v2 - u2 = 2aS III Equation of Motion