When we estimate distances from velocity data, it is...

When we estimate distances from velocity data, it is sometimes necessary to use times $t_0, t_1, t_2, t_3, \ldots$ that are not equally spaced. We can still estimate distances using the time periods $\Delta t_i=t_i-t_{i-1}$. For example, on May 7,1992 , the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. $$ \begin{array}{|l|c|c|} \hline \text { Event } & \text { Time (s) } & \text { Velocity (ft/s) } \\ \hline \text { Launch } & 0 & 0 \\ \text { Begin roll maneuver } & 10 & 185 \\ \text { End roll maneuver } & 15 & 319 \\ \text { Throttle to 89\% } & 20 & 447 \\ \text { Throttle to 67\% } & 32 & 742 \\ \text { Throttle to 104\% } & 59 & 1325 \\ \text { Maximum dynamic pressure } & 62 & 1445 \\ \text { Solid rocket booster separation } & 125 & 4151 \\ \hline \end{array} $$ Use these data to estimate the height above Earth's surface of the space shuttle Endeavour, 62 seconds after liftoff.

When we estimate distances from velocity data, it is sometimes necessary to use times $t_0, t_1, t_2, t_3, \ldots$ that are not equally spaced. We can still estimate distances using the time periods $\Delta t_i=t_i-t_{i-1}$. For example, on May 7,1992 , the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.
$$
\begin{array}{|l|c|c|}
\hline \text { Event } & \text { Time (s) } & \text { Velocity (ft/s) } \\
\hline \text { Launch } & 0 & 0 \\
\text { Begin roll maneuver } & 10 & 185 \\
\text { End roll maneuver } & 15 & 319 \\
\text { Throttle to 89\% } & 20 & 447 \\
\text { Throttle to 67\% } & 32 & 742 \\
\text { Throttle to 104\% } & 59 & 1325 \\
\text { Maximum dynamic pressure } & 62 & 1445 \\
\text { Solid rocket booster separation } & 125 & 4151 \\
\hline
\end{array}
$$
Use these data to estimate the height above Earth's surface of the space shuttle Endeavour, 62 seconds after liftoff.

Key Concepts

Kinematics of Motion

This concept refers to the relationship between velocity and displacement. In problems like estimating distance traveled, the displacement is determined by integrating the velocity function over time. Even when the time intervals are not equal, the integral sums the contributions from each time slice to yield the overall change in position.

Numerical Integration

Numerical integration involves approximating the integral of a function when an analytical solution is difficult or unnecessary. When data points come from measurements at discrete time intervals, numerical methods enable an estimation of the area under the velocity-time curve, which corresponds to the distance traveled.

Non-Uniform Time Intervals

In many real-world situations, data is collected at uneven time intervals. This requires adjustment in numerical integration techniques, ensuring that each time segment’s width is properly considered in estimating the integral. This approach accurately captures the cumulative effect of velocity over the varying intervals.

Trapezoidal Rule

The trapezoidal rule is a common method in numerical integration that approximates the area under a curve by dividing it into trapezoids. It works well when the data is given at discrete time points, even if the intervals are not equally spaced, making it a practical tool for estimating distance from velocity data.

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