3.4.1-eg-1 - (0 , ), so the only value of t in this...

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Example A particle moves according to the following law of motion s ( t ) = t + 4 t , t > 0 , where s is measured in meters and t in seconds. (a) What is the average velocity of the particle over the time interval [1 , 3]? (b) What is the (instantaneous) velocity of the particle at time t = 3 sec? (c) What is the acceleration of the particle at time t = 3 sec? (d) When is the particle moving in the positive direction, when is it moving in the negative direction, and when is it at rest? Solution: (a) The average velocity over the interval [1 , 3] is given by s (3) - s (1) 3 - 1 = ± 3 + 4 3 ² - ± 1 + 4 1 ² 2 m/sec. (b) The instantaneous velocity is s 0 ( t ), where s 0 ( t ) = 1 - 4 t 2 m/sec , so s 0 (3) = 1 - 4 3 2 = 9 - 4 9 = 5 9 m/sec . (c) Acceleration is given by s 00 ( t ) m/sec 2 , where s 00 ( t ) = ( - 2) ³ - 4 t 3 ´ = 8 t 3 m/sec 2 , so s 00 (3) = 8 3 3 = 8 27 m/sec 2 . (d) The particle is moving in the positive direction when s 0 ( t ) > 0, so we will first determine where s 0 ( t ) = 0. Combining the terms in s 0 ( t ), s 0 ( t ) = t 2 - 4 t 2 = 1 t 2 ± t 2 - 4 ² = 1 t 2 ( t - 2)( t + 2) , and since 1 /t 2 > 0 for all t > 0, we have that s 0 ( t ) = 0 , when ( t - 2)( t + 2) = 0 , or for t = 2 , - 2 sec . However, t = - 2 is not within the given time interval t
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Unformatted text preview: (0 , ), so the only value of t in this interval for which s ( t ) = 0 is t = 2, which means that s ( t ) 6 = 0 for t (0 , 2) and t (2 , ). To determine where s ( t ) is positive and negative, we can make a sign chart for the factors in s ( t ) when restricted to these two intervals: interval 1 t 2 t-2 t + 2 s ( t ) (0 , 2) +-+-(2 , ) + + + + The signs + or - in the table can determined by checking the value of the given factor at a single point in the interval. The conclusions to be drawn from the table are as follows: The particle is moving in the positive direction when the velocity satises s ( t ) > 0, or when t (2 , ). It is moving in the negative direction when the velocity satises s ( t ) < 0, or when t (0 , 2). It is at rest when s ( t ) = 0, or at t = 2 sec. 2...
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3.4.1-eg-1 - (0 , ), so the only value of t in this...

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