§4.1 Determining distance traveled from velocity (continued)

Summary

A biker's velocity data over four hours yields total distance traveled $40$ miles and net change in position $1$ mile north. These differ because

A toy rocket has vertical velocity $v(t) = 500 - 32t$ feet per second. Its velocity first equals zero at

The area of the triangle under $v(t) = 500 - 32t$ above the $t$-axis on $[0, 15.625]$ equals

A velocity graph consists of a triangle of area $\frac{1}{2}$ on $[0,1]$ followed by a quarter circle of radius $1$ on $[1,2]$, with later pieces below the axis. The exact distance traveled on $[0,2]$ is

The same graph encloses areas of about $1.285$ feet on $[0,2]$, $1.5$ feet on $[2,4]$, and a semicircle of radius $1$ on $[5,7]$. The position function

A pollution rate table reads $7, 8, 10, 13, 18, 35$ units per day at six day spacing over $30$ days. Since the rate is increasing, the left endpoint estimate $336$ units is

With $s(t) = 500t - 16t^2$ for the rocket, compute $s(5) - s(1)$ in feet.

For $v(t) = 500 - 32t$, give the time in seconds when the velocity reaches zero, rounded to three decimals.

For the same object, the signed areas on $[2,5]$ are $-\frac{1}{2}$, $-1$, and $-\frac{1}{2}$ across three one second pieces. Compute $s(5) - s(2)$ in feet.

For the biker, the total distance traveled being $40$ miles while the change in position is $1$ mile is perfectly consistent.