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5.13 Rate Measure (Distance, Velocity and Acceleration)

If y = s (t) represents the position function, then y = sí (t) represents the instantaneous

velocity i.e. and a = ví(t) = s" (t) represents the

instantaneous acceleration at time t i.e. a =

Note that a =

(1) If v > 0 (i.e. positive velocity) indicates that the position is increasing with increasing time while v < 0 (i.e. negative velocity) indicates the decreasing position as time decreases.

(2) If v = 0 implies that distance ísí remains constant on the given interval of time.

(3) If a > 0 implies that velocity is increasing with respect to time and a < 0 implies that the velocity is decreasing with respect to time.

(4) If v = constant on an interval of time, then a = 0 on that interval.

Example 36 A particle is moving in a straight line so that after ítí seconds its distance ísí from a fixed point O on the line is given by s (t) = 8t + t3.

Find (i) the velocity at time t and also t = 2

(ii) the initial velocity

(iii) acceleration at ítí and also at t = 2

Solution : s (t) = 8t + t3

(i) v = = 8 + 3t2

\ velocity at (t = 2) = 8 + 12 = 20 unit /s

(ii) Putting t = 0, initial velocity = 8 + 3 ī 0 = 8 unit /s

(iii) Acceleration a = = 6t

\ acceleration (at t = 2) = 6 ī 2 = 12 unit /s2

Example 37 The position of a particle on a line is a given by s (t) = 2t3 - 9t2 + 12t . Calculate its acceleration when it stops.

Solution : s (t) = 2t3 - 9t2 + 12t

\ v = = 6t2 - 18t + 12 and a = = 12t - 18

When the particle stops, its velocity (v) = 0

\ 6 (t2 - 3t + 2) = 0 Þ (t - 2) (t - 1) = 0

\ t = 1 and 2

When t = 1 a = 12 (1) - 18 = - 6 unit /s2 and

When t = 2 a = 12 (2) - 2 = 6 unit /s2


Example 38 A point moves in accordance with the law v (t) = a + bt + ct2 and the initial velocity and acceleration are 3 and 2 respectively. Also the acceleration at the end of 1st second is 12. Find the velocity at the end of 3 seconds and the acceleration at the end of 4 seconds.

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Index

5.1 Tangent And Normal Lines
5.2 Angle Between Two Curves
5.3 Interpretation Of The Sign Of The Derivative
5.4 Locality Increasing Or Decreasing Functions 5.5 Critical Points
5.6 Turning Points
5.7 Extreme Value Theorem
5.8 The Mean-value Theorem
5.9 First Derivative Test For Local Extrema
5.10 Second Derivative Test For Local Extrema
5.11 Stationary Points
5.12 Concavity And Points Of Inflection 5.13 Rate Measure (distance, Velocity And Acceleration)
5.14 Related Rates
5.15 Differentials : Errors And Approximation

Chapter 6





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