Answer:
0.01s : Average velocity = - 102.22 ft/s
0.005s : Average velocity = - 102.11 ft/s
0.002s : Average velocity = - 102.044 ft/s
0.001s: Average velocity = - 102.022 ft/s
Instantaneous velocity at t = 3 is - 102 ft/s
Step-by-step explanation:
The position-time function is expressed as
y = 30t - 22t^2
y(t) = 30t - 22t^2
1) To find the average velocity between t = 3 and t = 3 + 0.01 = 3.01, we would substitute these values into the position-time function. Thus,
y(3) = 30(3) - 22(3)^2 = - 108
y(3.01) = 30(3.01) - 22(3.01)^2 = - 109.0222
Average velocity = (- 109.0222 - - 108)/(3 - 3.01) = (- 109.0222 + 108)/0.01
Average velocity = - 102.22 ft/s
2) To find the average velocity between t = 3 and t = 3 + 0.005 = 3.005, we would substitute these values into the position-time function. Thus,
y(3) = 30(3) - 22(3)^2 = - 108
y(3.005) = 30(3.005) - 22(3.005)^2 = - 108.51055
Average velocity = (- 108.51055 - - 108)/(3 - 3.005) = (- 108.51055 + 108)/0.005
Average velocity = - 102.11 ft/s
3) To find the average velocity between t = 3 and t = 3 + 0.002 = 3.002, we would substitute these values into the position-time function. Thus,
y(3) = 30(3) - 22(3)^2 = - 108
y(3.002) = 30(3.002) - 22(3.002)^2 = - 108.204088
Average velocity = (- 108.204088 - - 108)/(3 - 3.005) = (- 108.204088 + 108)/0.002
Average velocity = - 102.044 ft/s
4) To find the average velocity between t = 3 and t = 3 + 0.001 = 3.001, we would substitute these values into the position-time function. Thus,
y(3) = 30(3) - 22(3)^2 = - 108
y(3.001) = 30(3.001) - 22(3.001)^2 = - 108.102022
Average velocity = (- 108.102022 - - 108)/(3 - 3.001) = (- 108.102022 + 108)/0.001
Average velocity = - 102.022 ft/s
To find the velocity function, we would differentiate the position time function. Thus,
v(t) = dy(t)/dt = 30 - 44t
To find the instantaneous velocity at t = 3, we would substitute t = 3 into v(t) = 30 - 44t
Thus,
v(3) = 30 - 44 x 3 = 30 - 132
v(3) = - 102
Instantaneous velocity at t = 3 is - 102 ft/s