If a ball is thrown into the air with a velocity of 48 ft/s, its height in feet t seconds later is given by y = 48t − 16t2.
(a) Find the average velocity for the time period beginning when t = 2 and lasting for each of the following.
(i) 0.5 seconds
ft/s
(ii) 0.1 seconds
ft/s
(iii) 0.05 seconds
ft/s
(iv) 0.01 seconds
ft/s
(b) Estimate the instantaneous velocity when t = 2.
ft/s
To find the average velocity for a given time period, you need to calculate the change in height divided by the change in time.
(a)
(i) To find the average velocity over 0.5 seconds, you need to calculate the change in height in the time interval from t = 2 to t = 2.5.
The initial height at t = 2 is y(2) = 48(2) - 16(2^2) = 96 - 64 = 32 feet.
The final height at t = 2.5 is y(2.5) = 48(2.5) - 16(2.5^2) = 120 - 100 = 20 feet.
Therefore, the change in height is 20 - 32 = -12 feet.
The change in time is 0.5 seconds.
The average velocity is the change in height divided by the change in time: -12 ft / 0.5 s = -24 ft/s.
(ii) To find the average velocity over 0.1 seconds, you need to calculate the change in height in the time interval from t = 2 to t = 2.1.
The initial height at t = 2 is y(2) = 32 feet.
The final height at t = 2.1 is y(2.1) = 48(2.1) - 16(2.1^2) = 100.8 - 92.88 = 7.92 feet.
Therefore, the change in height is 7.92 - 32 = -24.08 feet.
The change in time is 0.1 seconds.
The average velocity is the change in height divided by the change in time: -24.08 ft / 0.1 s = -240.8 ft/s.
(iii) To find the average velocity over 0.05 seconds, you need to calculate the change in height in the time interval from t = 2 to t = 2.05.
The initial height at t = 2 is y(2) = 32 feet.
The final height at t = 2.05 is y(2.05) = 48(2.05) - 16(2.05^2) = 98.4 - 85.12 = 13.28 feet.
Therefore, the change in height is 13.28 - 32 = -18.72 feet.
The change in time is 0.05 seconds.
The average velocity is the change in height divided by the change in time: -18.72 ft / 0.05 s = -374.4 ft/s.
(iv) To find the average velocity over 0.01 seconds, you need to calculate the change in height in the time interval from t = 2 to t = 2.01.
The initial height at t = 2 is y(2) = 32 feet.
The final height at t = 2.01 is y(2.01) = 48(2.01) - 16(2.01^2) = 96.48 - 86.8864 = 9.5936 feet.
Therefore, the change in height is 9.5936 - 32 = -22.4064 feet.
The change in time is 0.01 seconds.
The average velocity is the change in height divided by the change in time: -22.4064 ft / 0.01 s = -2240.64 ft/s.
(b) To estimate the instantaneous velocity when t = 2, you can calculate the average velocity over a smaller and smaller time interval centered around t = 2.
Let's estimate the instantaneous velocity over a 0.001-second time interval centered around t = 2.
The initial height at t = 2 is y(2) = 32 feet.
The final height at t = 2.001 is y(2.001) = 48(2.001) - 16(2.001^2) = 96.048 - 86.413536 = 9.634464 feet.
Therefore, the change in height is 9.634464 - 32 = -22.365536 feet.
The change in time is 0.001 seconds.
The average velocity is the change in height divided by the change in time: -22.365536 ft / 0.001 s = -22,365.536 ft/s.
This is an estimate of the instantaneous velocity when t = 2.