A particle moves along a straight line with continuous velocity given in meters per second

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A particle moves along a straight line with continuous velocity

v

given in meters per second. On the time interval

0\le t\le 6

, the velocity satisfies

v(t)=\sqrt{9+t^{2}}

. For a regular partition of the interval into

n

subintervals, let

t_i=\dfrac{6i}{n}

and take the mid-point

\bar t_i=\dfrac{t_{i-1}+t_i}{2}

of each subinterval. (a) Write an expression involving a limit of Riemann sums that gives the exact displacement of the particle on

0\le t\le 6

. (b) Rewrite the limit from part (a) as a single definite integral. (c) Evaluate the integral found in part (b) by making an appropriate substitution. Give the exact numerical value of the displacement in meters. (d) The average velocity of the particle on

0\le t\le 6

is defined to be

\dfrac{1}{6}

times the displacement. Determine the average velocity and indicate the appropriate units. (e) Define an accumulation function

s(x)=\displaystyle \int_{0}^{x} \sqrt{9+t^{2}}\,dt

for

x\ge 0

. Compute

s'(x)

and interpret its value in the context of the particle’s motion.

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