Find (a) Displacement (b) Distance Traveled

In Exercises 3386, the velncity functinn, in feet per secnntl, is given for a particle musing along a straight line. Find (a) the displacement and (h) the total distance that the particle travels user the given intervaL 83.v{r}=5rT, 05:53 84.v{r}=r2r12,15r55 85. v{r}=r3 1013+2'H IS, 15 r 5? 86. v{r)=r38:2+15:, 05:55 When calculating the Iota! distance traveled by the particle, you must consider the intervals where v(r) E 0 and the intervals where v(r} 2 0. When v(r} E 0, the particle moves to the left, and when v(r} a 0, the particle moves to the right. To calculate the total distance traveled, integrate the absolute value of velocity Iv[r} I So, the displacement of a particle and the total distance traveled by a particle over [(1, b] can be written as b Displacement on [11, b] = I 11$)ch = A] A2 + A3 a b Total distance traveled on [11, b] = I |v(r)| d: = A1 + A2 + A, .' A particle is moving along a line so that its velocity is v(r) = I3 - 1041?2 + 29! - 2'3 feet per second at time I. a. What is the displacement of the particle on the time interval 1 E I S 5'? b. What is the total distance traveled by the particle on the time interval 1 E t E 5'? Solution :1. By definition, you know that the displacement is 5 5 J- v(:) dr = J (r3 10:2 + 29: 20) d: l l _ _s 2_ T |:4 3I+2I 2011 = E _ (-) 12 12 _ E ' 12 :2 3. So, the particle moves feet to the right