If a ball is throw into the air with a velocity of 40ft/s its height in feet t seconds later is given by: y=40t-16t^2. Find the average velocity for the time period beginning when t=2 and lasting 0.5 seconds.
I assume the ball is thrown straight up, so the velocity only has 1 dimensional components.
The average velocity is the same as the average slope over the given time period.
m = (y1 - y2) / (x1 - x2)
average V = [y(2) - y(2.5)] / (2 - 2.5)
ohh i see it now! thank you!
To find the average velocity for the given time period, we can use the formula for average velocity:
Average velocity = (change in position) / (change in time)
In this case, the position of the ball is given by the equation y = 40t - 16t^2.
Let's find the initial and final positions at t = 2 and t = 2.5 seconds, respectively.
At t = 2 seconds:
y1 = 40(2) - 16(2)^2
y1 = 80 - 16(4)
y1 = 80 - 64
y1 = 16 feet
At t = 2.5 seconds:
y2 = 40(2.5) - 16(2.5)^2
y2 = 100 - 16(6.25)
y2 = 100 - 100
y2 = 0 feet
Now, let's calculate the change in position:
Change in position = y2 - y1
Change in position = 0 - 16
Change in position = -16 feet
The change in time is given as 0.5 seconds.
Now we can substitute the values into the formula for average velocity:
Average velocity = (change in position) / (change in time)
Average velocity = (-16) / (0.5)
Average velocity = -32 feet per second (ft/s)
Therefore, the average velocity for the time period beginning when t = 2 and lasting 0.5 seconds is -32 ft/s.
To find the average velocity over a specific time period, we need to find the total displacement divided by the total time.
In this case, the height of the ball at a given time t is given by the equation y = 40t - 16t^2.
To find the average velocity for the time period beginning when t=2 and lasting 0.5 seconds, we need to find the height at t=2 and t=2.5 seconds, and then calculate the change in height and divide it by the change in time.
First, let's calculate the height at t=2 seconds:
y = 40t - 16t^2
y = 40(2) - 16(2)^2
y = 80 - 16(4)
y = 80 - 64
y = 16 feet
Next, let's calculate the height at t=2.5 seconds:
y = 40t - 16t^2
y = 40(2.5) - 16(2.5)^2
y = 100 - 16(6.25)
y = 100 - 100
y = 0 feet
Now we can calculate the change in height and the change in time:
Change in height = Final height - Initial height
Change in height = 0 - 16
Change in height = -16 feet
Change in time = Final time - Initial time
Change in time = 2.5 - 2
Change in time = 0.5 seconds
Finally, we can calculate the average velocity using the formula:
Average velocity = Change in height / Change in time
Average velocity = (-16 feet) / (0.5 seconds)
Average velocity = -32 feet/second
Therefore, the average velocity for the time period beginning when t=2 and lasting 0.5 seconds is -32 feet/second.