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Motion III
Velocity/Time graph
Uniform Motion The Velocity /Time graph of an object moving in uniform motion is a straight line which passes through the origin.
Slope = y_{2}y_{1}/x_{2}x_{1 }Where y is taken from the distance and y_{2}y_{1 }
Where x is taken from the time and x_{2}x_{1 }
Zero Acceleration The graph of an object with zero acceleration is a straight line which is parallel to the x axis
Uniform Retardation The graph of an object with uniform retardation is a straight line and the line meets the x axis when the object comes to rest.
Variable Retardation/Acceleration The graph of an object with Variable Retardation/Acceleration is a curve line. The slope cannot be calculated.
Graphical Derivations of Equations 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 velocitytime 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 1^{st} 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 equation of motion; v  u = at)
2^{nd} Equation of motion
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 a^{2}  b^{2} = (a+b) (ab)]
v^{2}  u^{2} = 2aS 3^{rd} Equation of Motion
Circular Motion/Accelerated Motion
When an object moves along a circular path with constant speed its direction keeps on changing continuously hence it is known as acceleration motion. At any instant the direction of motion is along the tangent to the circle at that point.
