A person standing on the roof of a building throws a ball directly upward. The ball misses the rooftop on its
way down and eventually strikes the ground. The function s(t) = −16t2 + 64t + 80 describes the ball’s height
above the ground, s(t) , in feet, t , seconds after it was thrown.
a. Find the ball’s average velocity between the time it was thrown and 2 seconds later.
b. Find the ball’s average velocity between 2 and 4 seconds after it was thrown.
Since v(t) = 64 - 32t
v(2) = 0
So, the velocity decreases from 64 to 0 in 2 seconds. Average v = 32
Similarly, once the ball starts falling, it regains its velocity of -64 after 2 seconds. So, on the way down, avg v = -32
a. To find the ball's average velocity between the time it was thrown and 2 seconds later, we need to calculate the change in position (height) divided by the change in time.
The time it was thrown is 0 seconds, and 2 seconds later is at t = 2.
To find the change in position, we subtract the height at 0 seconds from the height at 2 seconds:
s(2) - s(0) = (-16(2)^2 + 64(2) + 80) - (-16(0)^2 + 64(0) + 80)
= (-64 + 128 + 80) - (0 + 0 + 80)
= 144 feet
To find the change in time, we subtract the initial time (0 seconds) from the final time (2 seconds):
2 - 0 = 2 seconds
Now, we can calculate the average velocity:
Average Velocity = Change in position / Change in time
= 144 feet / 2 seconds
= 72 feet per second
Therefore, the ball's average velocity between the time it was thrown and 2 seconds later is 72 feet per second.
b. To find the ball's average velocity between 2 and 4 seconds after it was thrown, we follow the same process as in part a.
The initial time is t = 2 seconds, and 4 seconds later is at t = 4.
To find the change in position, we subtract the height at 2 seconds from the height at 4 seconds:
s(4) - s(2) = (-16(4)^2 + 64(4) + 80) - (-16(2)^2 + 64(2) + 80)
= (-256 + 256 + 80) - (-64 + 128 + 80)
= 80 feet
To find the change in time, we subtract the initial time (2 seconds) from the final time (4 seconds):
4 - 2 = 2 seconds
Now, we can calculate the average velocity:
Average Velocity = Change in position / Change in time
= 80 feet / 2 seconds
= 40 feet per second
Therefore, the ball's average velocity between 2 and 4 seconds after it was thrown is 40 feet per second.
To find the ball's average velocity, we need to calculate the change in height divided by the change in time within the given time interval.
a. To find the ball's average velocity between the time it was thrown and 2 seconds later, we need to calculate the change in height and the change in time.
The time interval is from the moment the ball was thrown (t = 0) until 2 seconds later (t = 2).
Let's calculate the average velocity:
1. Find the height at the start of the time interval (t = 0):
s(0) = -16(0)^2 + 64(0) + 80
s(0) = 80 feet
2. Find the height at the end of the time interval (t = 2):
s(2) = -16(2)^2 + 64(2) + 80
s(2) = -64 + 128 + 80
s(2) = 144 feet
3. Calculate the change in height:
Change in height = s(2) - s(0)
Change in height = 144 - 80
Change in height = 64 feet
4. Calculate the change in time:
Change in time = 2 - 0
Change in time = 2 seconds
5. Calculate the average velocity:
Average velocity = Change in height / Change in time
Average velocity = 64 feet / 2 seconds
Average velocity = 32 feet/second
Therefore, the ball's average velocity between the time it was thrown and 2 seconds later is 32 feet/second.
b. To find the ball's average velocity between 2 and 4 seconds after it was thrown, we follow the same process:
1. Find the height at the start of the time interval (t = 2):
s(2) = 144 feet (from previous calculation)
2. Find the height at the end of the time interval (t = 4):
s(4) = -16(4)^2 + 64(4) + 80
s(4) = -256 + 256 + 80
s(4) = 80 feet
3. Calculate the change in height:
Change in height = s(4) - s(2)
Change in height = 80 - 144
Change in height = -64 feet
4. Calculate the change in time:
Change in time = 4 - 2
Change in time = 2 seconds
5. Calculate the average velocity:
Average velocity = Change in height / Change in time
Average velocity = -64 feet / 2 seconds
Average velocity = -32 feet/second
Therefore, the ball's average velocity between 2 and 4 seconds after it was thrown is -32 feet/second.