A particle moves in a straight line from its initial position so that after t seconds, it distance is given by s = t^2 + t feet from its initial position.

Question

(a) Find the average velocity of the particle over the interval [1; 3] seconds.
(b) Find the instantaneous velocity of the particle at t = 1 second.

Answer 1

s(t) = t² + t

 

a)  s(1) = 1²+1 = 2, s(3) = 3²+3 = 12.  So the end points of the interval are (1,2) and (3,12).  The average velocity in this interval is the slope of the secant line connecting the two points:

 

average velocity = (12-2)/(3-1) = 5 ft/sec

 

b) The instantaneous velocity is the derivative of s wrt t:

 

instantaneous velocity = ds/dt = d(t²+t)/dt = 2t+1

 

At t = 1:

 

instantaneous velocity = 2*1 + 1= 3 ft/sec