A ball is thrown vertically upward with an initial velocity of 128 feet per second, the ball’s height after t second is s(t) = 128t – 16t².

a. Find the average velocity from 2 to 4 seconds.

V = Vo + g*t = 128 - 32*2 = 64 Ft/s after 2 s.

V = 128 - 32*4 = 0 After 4 s.

Vavg = (0+4)/2 = 2 Ft/s

To find the average velocity from 2 to 4 seconds, we need to find the average rate of change of the height of the ball over that time interval.

The average velocity is calculated by taking the difference in height between the initial and final time, and dividing it by the difference in time:

Average Velocity = (Change in height) / (Change in time)

In this case, the initial time is 2 seconds and the final time is 4 seconds, so the change in time is 4 - 2 = 2 seconds.

Now we need to find the change in height over that time interval. To do this, we plug in the initial and final time into the height equation and subtract the results:

s(4) = 128(4) - 16(4)² = 512 - 256 = 256 feet

s(2) = 128(2) - 16(2)² = 256 - 64 = 192 feet

Change in height = s(4) - s(2) = 256 - 192 = 64 feet

Finally, we can calculate the average velocity by dividing the change in height by the change in time:

Average Velocity = Change in height / Change in time = 64 feet / 2 seconds = 32 feet per second.

So, the average velocity from 2 to 4 seconds is 32 feet per second.

To find the average velocity from 2 to 4 seconds, we need to find the displacement of the ball during that time period and divide it by the time taken.

First, let's find the height of the ball at 2 seconds and 4 seconds using the given equation s(t) = 128t - 16t^2.

At t = 2 seconds:
s(2) = 128(2) - 16(2)^2
= 256 - 16(4)
= 256 - 64
= 192 feet

At t = 4 seconds:
s(4) = 128(4) - 16(4)^2
= 512 - 16(16)
= 512 - 256
= 256 feet

Next, let's find the displacement by subtracting the initial height from the final height:
Displacement = s(4) - s(2)
= 256 - 192
= 64 feet

Finally, we can find the average velocity by dividing the displacement by the time taken:
Average velocity = Displacement / Time
= 64 feet / (4 seconds - 2 seconds)
= 64 feet / 2 seconds
= 32 feet per second

Therefore, the average velocity from 2 to 4 seconds is 32 feet per second.