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Kamis, 02 Juni 2011

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Equations of Motion

The variable quantities in a uniformly accelerated rectilinear motion are time, speed, distance covered and acceleration. Simple relations exist between these quantities. These relations are expressed in terms of equations called equations of motion
The equations of motion are:
(1) v = u + at

three equation of motion
(3) v2 - u2 = 2aS

Derivation of the First Equation of Motion

Consider a particle moving along a straight line with uniform acceleration 'a'. At t = 0, let the particle be at A and u be its initial velocity and when t = t, v be its final velocity.
formula to calculate acceleration
v = u + at  I equation of motion

Second Equation of Motion

From equations (1) and (2)
The first equation of motion is v = u + at.
Substituting the value of v in equation (3), we get
II equation of motion

Third Equation of Motion

The first equation of motion is v = u + at.
v - u = at ... (1)
From equation (2) and equation (3) we get,
Multiplying equation (1) and equation (4) we get,

III equation of motion

(v - u) (v + u) = 2aS
[We make use of 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 representation of I 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 =

I equation of motion

v - u = at
v = u + at  I equation of motion

Second Equation of Motion

Let u be the initial velocity of an object and 'a' the acceleration produced in the body. The distance travelled S in time t is given by the area enclosed by the velocity-time graph for the time interval 0 to t.
graphical representation of II 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)
II equation of motion

Third Equation of Motion

Let 'u' be the initial velocity of an object and a be the acceleration produced in the body. The distance travelled 'S' in time 't' is given by the area enclosed by the v - t graph.
graphical representation of III equation of motion
Graphical Derivation of Third Equation
S = area of the trapezium OABD.
Substituting the value of t in equation (1) we get,
III equation of motion
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

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