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 Δti = 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 = ftEvent Time (s) Velocity (ft/s)Launch 0 0Begin roll maneuver 10 180End roll maneuver 15 319Throttle to 89% 20 453Throttle to 67% 32 742Throttle to 104% 59 1100Maximum dynamic pressure 62 1430Solid rocket booster separation 125 4052
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 Δti = 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 = ftEvent Time (s) Velocity (ft/s)Launch 0 0Begin roll maneuver 10 180End roll maneuver 15 319Throttle to 89% 20 453Throttle to 67% 32 742Throttle to 104% 59 1100Maximum dynamic pressure 62 1430Solid rocket booster separation 125 4052
Solution
To estimate the height of the space shuttle 62 seconds after liftoff, we can use the trapezoidal rule for approximation. The trapezoidal rule is a method for approximating the definite integral of a function. In this case, we are integrating velocity over time to get distance (or height in this case).
The trapezoidal rule states that the integral of a function f(x) from a to b is approximately equal to (b - a) * [f(a) + f(b)] / 2.
We can apply this rule to each time interval in the data given:
- From t0 to t1 (0 to 10 seconds): (10 - 0) * [0 + 180] / 2 = 900 ft
- From t1 to t2 (10 to 15 seconds): (15 - 10) * [180 + 319] / 2 = 1247.5 ft
- From t2 to t3 (15 to 20 seconds): (20 - 15) * [319 + 453] / 2 = 1926 ft
- From t3 to t4 (20 to 32 seconds): (32 - 20) * [453 + 742] / 2 = 11950 ft
- From t4 to t5 (32 to 59 seconds): (59 - 32) * [742 + 1100] / 2 = 40714 ft
- From t5 to t6 (59 to 62 seconds): (62 - 59) * [1100 + 1430] / 2 = 3825 ft
Adding these up, we get the total height of the space shuttle 62 seconds after liftoff:
900 + 1247.5 + 1926 + 11950 + 40714 + 3825 = 59562.5 ft
So, the space shuttle was approximately 59562.5 feet above Earth's surface 62 seconds after liftoff.
Similar Questions
EXAMPLE 4 Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30 second time interval. We take the speedometer readings every five seconds and record them in the following table.Time (s) 0 5 10 15 20 25 30Velocity (mi/h) 17 19 25 27 33 30 28In order to have the time and the velocity in consistent units, let's convert the velocity readings to feet per second (1 mi/h = 52803600 ft/s). (Round your answers to the nearest whole number.)Time (s) 0 5 10 15 20 25 30Velocity (ft/s) 25 37 48 44 41
The relation between time ' t ' and distance ' x ' is t= αx2+βx, where α and β are constants. The relation between acceleration (a) and velocity (v) is:
Calculate the time taken for Bolt to reach each distance in the race according to the data in the table above.Distance (metres)020406080100Time taken (seconds)
Shwetha is fascinated by physics and wants to calculate the distance covered by an object in motion. She has the initial speed, time, and acceleration values. The program takes these values as input, calculates the distance using the formula (distance = initial speed * time + 0.5 * acceleration * time2), and displays the result in meters.
The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. ft (smaller value) ft (larger value)t (s) 0 0.5 1.0 1.5 2.0 2.5 3.0v (ft/s) 0 6.7 9.2 15.5 17.5 19.8 20.2
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.