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Initial acceleration of a particle moving in a straight line is a0 and initial velocity is zero. The acceleration reduces continuously to half in every t0 seconds. The terminal speed of the particle is:
The final velocity of the particle for the first t seconds is given as v 1 =at, for the next t seconds is, v 2 = a t + a t / 2 and so on. So yes, you can apply the infinite geometric sum for these.
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