If a ball is thrown in the air with a velocity 44 ft/s, its height in feet t seconds later is given by y = 44t - 16t2.
(a) Find the average velocity for the time period beginning when t = 2 and lasting 0.5 second.
ft/s
(b) Find the average velocity for the time period beginning when t = 2 and lasting 0.1 second.
ft/s
(c) Find the average velocity for the time period beginning when t = 2 and lasting 0.05 second.
ft/s
(d) Find the average velocity for the time period beginning when t = 2 and lasting 0.01 second.
? ft/s
(e) Estimate the instantaneous velocity when t = 2.
This will not use calculus. It is an exercise to show how to use algebra to approximate the instantaneous velocity that you would get using calculus.
Just use the formula
y = 44t - 16 t^2
multiple times.
For (a), y(2) = 88 - 64 = 24 ft
y(2.5) = 110 - 100 = 10 ft
Average velocity Vav
= [y(2.5) - y(2)]/0.5 = -28 ft/s
For (d), y(2.01)= 88.440 - 64.6416 = 23.7984
Vav = [y(2.01) - y(2)]/0.01
= (23.7984 -24)/0.01 = -20.16 ft/s
Do parts (b) and (c) yourself and notice the trends in average velocity as the time interval gets shorter
how do you solve for (e)
To find the average velocity, we need to calculate the change in height divided by the change in time.
(a) Average velocity for the time period beginning when t = 2 and lasting 0.5 seconds:
Change in height = y(2 + 0.5) - y(2) = (44*(2+0.5) - 16*(2+0.5)^2) - (44*2 - 16*2^2)
Average velocity = Change in height / Change in time = (44*(2+0.5) - 16*(2+0.5)^2) - (44*2 - 16*2^2) / 0.5
Simplifying the equation will give the answer in ft/s.
(b) Average velocity for the time period beginning when t = 2 and lasting 0.1 seconds:
Similar to part (a), calculate the change in height for 0.1 seconds and divide it by 0.1 to get the answer in ft/s.
(c) Average velocity for the time period beginning when t = 2 and lasting 0.05 seconds:
Repeat the same steps as before, but now calculate the change in height for 0.05 seconds and divide it by 0.05 to get the answer in ft/s.
(d) Average velocity for the time period beginning when t = 2 and lasting 0.01 seconds:
Repeat the same steps as before, but now calculate the change in height for 0.01 seconds and divide it by 0.01 to get the answer in ft/s.
(e) To estimate the instantaneous velocity when t = 2, you need to calculate the derivative of the height function y with respect to time t and evaluate it at t = 2. This will give you the instantaneous velocity at that moment.
To find the average velocity for a given time period, we need to calculate the change in position (height) divided by the change in time within that period.
(a) To find the average velocity for a time period of 0.5 seconds starting at t = 2, we can substitute the given values into the equation for height: y = 44t - 16t^2.
First, calculate the height at t = 2. Plug in t = 2 into the equation: y = 44(2) - 16(2)^2. Simplify: y = 88 - 64 = 24 ft.
Next, calculate the height at t = 2.5 (0.5 seconds after t = 2): y = 44(2.5) - 16(2.5)^2. Simplify: y = 110 - 100 = 10 ft.
Now, calculate the change in position (height) over this time period: Δy = 10 - 24 = -14 ft.
Finally, divide the change in position by the change in time to find the average velocity: Average velocity = Δy / Δt = -14 ft / 0.5 s = -28 ft/s.
Therefore, the average velocity for the time period beginning when t = 2 and lasting 0.5 second is -28 ft/s.
(b) To find the average velocity for a time period of 0.1 seconds starting at t = 2, follow the same process as above. Calculate the height at t = 2 and t = 2.1, find the change in position (height), and then divide by the change in time.
(c) To find the average velocity for a time period of 0.05 seconds starting at t = 2, follow the same process as above. Calculate the height at t = 2 and t = 2.05, find the change in position (height), and then divide by the change in time.
(d) To find the average velocity for a time period of 0.01 seconds starting at t = 2, follow the same process as above. Calculate the height at t = 2 and t = 2.01, find the change in position (height), and then divide by the change in time.
(e) The instantaneous velocity refers to the velocity at a specific point in time. In this case, we need to find the derivative of the equation for height with respect to time. The derivative of y = 44t - 16t^2 is dy/dt = 44 - 32t.
Evaluate the derivative at t = 2: dy/dt = 44 - 32(2) = 44 - 64 = -20 ft/s.
Therefore, the estimated instantaneous velocity when t = 2 is -20 ft/s.