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| Relations for Uniformly Accelerated Motion (Graphical Treatment) |
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| The velocity-time graph for uniformly accelerated rectilinear motion (motion along a straight line), a > 0, is illustrated in the figure. |
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| The velocity-time graph for a uniformly retarded motion is as shown in the figure. Here a < 0, indicating retardation or deceleration. |
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| In this section, we are going to derive the velocity-time relation for a particle moving with constant acceleration. |
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| The figure above illustrates a particle moving with constant acceleration 'a' along the positive direction of X-axis. If v (t1) and v (t2) be the velocities of the particle at times t1 and t2 respectively, then |
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Acceleration,  according
to the definition |
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| Cross-multiplying, we have, |
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| v (t2) - v (t1) = a (t2 - t1) |
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 v (t2) = v (t1) + a (t2 - t1) |
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| If the motion is considered from t = 0, then the corresponding velocity v(0) at t = 0 is called the initial velocity and the velocity after a time interval, say t, is called the final velocity and denoted by v(t). In this case, the acceleration is given by |
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| or at = v (t) - v (0) |
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 v(t) = v (0) + at |
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| The above expression represents the first equation of motion. |
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| The figure below illustrates the velocity-time graph AB of a uniformly accelerated particle. The points A and B correspond to velocities v (0) and v (t) respectively. |
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| Slope of the straight line AB |
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| Also, from the definition of acceleration, we know that |
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 We conclude that the slope of the velocity-time graph for uniformly accelerated motion gives the acceleration. |
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| At any given time, the direction of motion is given by velocity and not by acceleration. As an example, when a ball is thrown up vertically, its velocity is directed upwards at any given time, but its acceleration is always directed in the downward direction. |
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