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| Equations of Motion |
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| 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 |
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| The equations of motion are: |
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| (1) v = u + at |
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| (3) v2 - u2 = 2aS |
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| 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. |
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| v = u + at I equation of motion |
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| From equations (1) and (2) |
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| The first equation of motion is v = u + at. |
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| Substituting the value of v in equation (3), we get |
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| The first equation of motion is v = u + at. |
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| v - u = at ... (1) |
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| From equation (2) and equation (3) we get, |
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| Multiplying equation (1) and equation (4) we get, |
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| (v - u) (v + u) = 2aS |
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| [We make use of the identity a2 - b2 = (a + b) (a - b)] |
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| v2 - u2 = 2aS III equation of motion |
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| First Equation of Motion |
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| Graphical Derivation of First Equation |
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| 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. |
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| The figure shows the velocity-time graph of the motion of the object. |
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| Slope of the v - t graph gives the acceleration of the moving object. |
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Thus, acceleration = slope = AB =  |
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| v - u = at |
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| v = u + at I equation of motion |
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| Second Equation of Motion |
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| 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. |
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| Graphical Derivation of Second Equation |
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| Distance travelled S = area of the trapezium ABDO |
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| = area of rectangle ACDO + area of DABC |
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| (v = u + at I eqn of motion; v - u = at) |
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| Third Equation of Motion |
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| 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. |
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| Graphical Derivation of Third Equation |
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| S = area of the trapezium OABD. |
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| Substituting the value of t in equation (1) we get, |
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| 2aS = (v + u) (v - u) |
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| (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] |
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| v2 - u2 = 2aS III Equation of Motion |
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