A ride in an amusement park drops the riders from a height of 530 feet. The height (in feet) of the riders [tex]$t$[/tex] seconds after freefall is given by:
[tex]$f(t)=-16 t^2+530$[/tex]
a. What is the average velocity of the rider between [tex]$t=1$[/tex] second to [tex]$t=6$[/tex] seconds after the freefall?
Average Velocity: $\square$
b. Complete a table to approximate the instantaneous velocity of the rider exactly [tex]$t=1$[/tex] second after the freefall.
Instantaneous velocity: $\square$
Answer (1)
Calculate the average velocity between t = 1 and t = 6 using the formula 6 − 1 f ( 6 ) − f ( 1 ) , which equals 5 − 46 − 514 = − 112 feet per second.
Approximate the instantaneous velocity at t = 1 by calculating average velocities over small intervals h = 0.1 , 0.01 , 0.001 .
For h = 0.1 , the average velocity is 0.1 f ( 1.1 ) − f ( 1 ) = − 33.6 feet per second; for h = 0.01 , it's − 32.16 feet per second; and for h = 0.001 , it's − 32.016 feet per second.
The instantaneous velocity at t = 1 is approximately − 32 feet per second.
Explanation
Problem Analysis We are given the height function f ( t ) = − 16 t 2 + 530 , which describes the height of a rider t seconds after a freefall from 530 feet. We need to find the average velocity between t = 1 and t = 6 seconds, and then approximate the instantaneous velocity at t = 1 second.
Calculating Average Velocity To find the average velocity between t = 1 and t = 6 , we use the formula: Average velocity = 6 − 1 f ( 6 ) − f ( 1 ) First, we calculate f ( 1 ) and f ( 6 ) :
f ( 1 ) = − 16 ( 1 ) 2 + 530 = − 16 + 530 = 514 f ( 6 ) = − 16 ( 6 ) 2 + 530 = − 16 ( 36 ) + 530 = − 576 + 530 = − 46 Now, we plug these values into the average velocity formula: Average velocity = 6 − 1 − 46 − 514 = 5 − 560 = − 112 The average velocity of the rider between t = 1 and t = 6 seconds is -112 feet per second.
Approximating Instantaneous Velocity To approximate the instantaneous velocity at t = 1 , we can calculate the average velocity over very small intervals around t = 1 . We'll use the formula: Average velocity = h f ( 1 + h ) − f ( 1 ) where h is a small change in time. We will use h = 0.1 , 0.01 , and 0.001 to get increasingly accurate approximations. For h = 0.1 :
f ( 1 + 0.1 ) = f ( 1.1 ) = − 16 ( 1.1 ) 2 + 530 = − 16 ( 1.21 ) + 530 = − 19.36 + 530 = 510.64 Average velocity = 0.1 510.64 − 514 = 0.1 − 3.36 = − 33.6 For h = 0.01 :
f ( 1 + 0.01 ) = f ( 1.01 ) = − 16 ( 1.01 ) 2 + 530 = − 16 ( 1.0201 ) + 530 = − 16.3216 + 530 = 513.6784 Average velocity = 0.01 513.6784 − 514 = 0.01 − 0.3216 = − 32.16 For h = 0.001 :
f ( 1 + 0.001 ) = f ( 1.001 ) = − 16 ( 1.001 ) 2 + 530 = − 16 ( 1.002001 ) + 530 = − 16.032016 + 530 = 513.967984 Average velocity = 0.001 513.967984 − 514 = 0.001 − 0.032016 = − 32.016 As h approaches 0, the average velocity approaches -32.
Final Answer Based on the calculations above, we can fill in the answers: a. Average Velocity: -112 feet per second. b. Instantaneous velocity: Approximately -32 feet per second.
Examples
Understanding average and instantaneous velocity is crucial in many real-world scenarios. For example, when designing roller coasters, engineers need to calculate the average velocity of the cars between different points to ensure the ride is thrilling but safe. They also need to know the instantaneous velocity at specific points, like the bottom of a drop, to design the track and braking systems effectively. Similarly, in automotive engineering, understanding these concepts helps in designing safer and more efficient vehicles.
