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.Find the average velocity from 2 to 4 seconds.
avg velocity is (total distance)/(total time)
so, that would be
(s(4)-s(2))/(4-2)
derivative V:
V (average) = dx/dt = 128 - 16 x 2 t^2-1
= 128 - 32t
substitute the time given
t = 4:
128 - 32 x 4 = 0
t=0
t = 2:
128 - 32 x 2 = 64
t=64
To find the average velocity from 2 to 4 seconds, we need to calculate the change in height and the change in time during this interval.
First, let's find the height at t = 2 and t = 4 seconds using the given equation:
For t = 2 seconds:
s(2) = 128(2) - 16(2)²
s(2) = 256 - 16(4)
s(2) = 256 - 64
s(2) = 192 feet
For t = 4 seconds:
s(4) = 128(4) - 16(4)²
s(4) = 512 - 16(16)
s(4) = 512 - 256
s(4) = 256 feet
Next, let's calculate the change in height:
Change in height = s(4) - s(2)
Change in height = 256 - 192
Change in height = 64 feet
Now, let's calculate the change in time:
Change in time = 4 seconds - 2 seconds
Change in time = 2 seconds
Finally, let's find the average velocity:
Average velocity = Change in height / Change in time
Average velocity = 64 feet / 2 seconds
Average velocity = 32 feet per second
Therefore, the average velocity from 2 to 4 seconds is 32 feet per second.
To find the average velocity of the ball from 2 to 4 seconds, we need to find the displacement of the ball during that time period and divide it by the time interval. The displacement can be found by subtracting the initial position from the final position.
First, let's find the initial and final positions of the ball at 2 seconds and 4 seconds, respectively.
For t = 2 seconds:
s(2) = 128(2) - 16(2)²
s(2) = 256 - 16(4)
s(2) = 256 - 64
s(2) = 192 feet
For t = 4 seconds:
s(4) = 128(4) - 16(4)²
s(4) = 512 - 16(16)
s(4) = 512 - 256
s(4) = 256 feet
The initial position at t = 2 seconds is 192 feet, and the final position at t = 4 seconds is 256 feet.
Now, let's calculate the displacement:
Displacement = Final position - Initial position
Displacement = 256 feet - 192 feet
Displacement = 64 feet
Next, we need to find the time interval:
Time interval = Final time - Initial time
Time interval = 4 seconds - 2 seconds
Time interval = 2 seconds
Finally, we can calculate the average velocity by dividing the displacement by the time interval:
Average velocity = Displacement / Time interval
Average velocity = 64 feet / 2 seconds
Average velocity = 32 feet per second
Therefore, the average velocity of the ball from 2 to 4 seconds is 32 feet per second.