When we estimate distances from velocity data, it is sometimes necessary to use

Question

When we estimate distances from velocity data, it is sometimes necessary to use times t0, t1, t2, t3, . . . that are not equally spaced.  We can still estimate distances using the time periods ?ti = ti ? ti ? 1. For example, a space shuttle was launched on a mission, the purpose of which was to install a new motor in a satellite. The table provided gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height, h, above Earth's surface of the space shuttle, 62 seconds after liftoff. (Give the upper approximation available from the data.)
h = 1      ft Event Time (s) Velocity (ft/s) Launch 0      0 Begin roll maneuver 10      180 End roll maneuver 15      319 Throttle to 89% 20      442 Throttle to 67% 32      742 Throttle to 104% 59      1217 Maximum dynamic pressure 62      1430 Solid rocket booster separation 125      4052 t0, t1, t2, t3, . . . ?ti = ti ? ti ? 1. Event Time (s) Velocity (ft/s) Launch 0      0 Begin roll maneuver 10      180 End roll maneuver 15      319 Throttle to 89% 20      442 Throttle to 67% 32      742 Throttle to 104% 59      1217 Maximum dynamic pressure 62      1430 Solid rocket booster separation 125      4052 Event Time (s) Velocity (ft/s) Launch 0      0 Begin roll maneuver 10      180 End roll maneuver 15      319 Throttle to 89% 20      442 Throttle to 67% 32      742 Throttle to 104% 59      1217 Maximum dynamic pressure 62      1430 Solid rocket booster separation 125      4052

Explanation / Answer

h ? 10 [ 180] + 5 [ 319 ] + 5 [ 453 ] + 12 [ 742 ] + 27 [ 1325 ] + 3 [ 1453 ]

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