If a rock is thrown upward on the planet Mars with a velocity of 7 m/s, its height in meters t seconds later is given by y = 7t − 1.86t2. (Round your answers to two decimal places.)
(a) Find the average velocity over the given time intervals.
(i) [1, 2]
(ii) [1, 1.5]
(iii)[1, 1.1]
(iv) [1, 1.01]
(v) [1, 1.001]
(b) Estimate the instantaneous velocity when t = 1.
the equation is y= 7t-1.86t^2
To get the average velocity (m/s), on the interval [a,b] just calculate [y(b)-y(a)]/(b-a) -- in other words, how far it moved (distance), divided by how long it took (time)
I figure you can probably do that. You are, after all, taking calculus, so your algebra I should be OK.
The results in steps i-v should show a sequence converging on y'(1) where
y'(t) = 7 - 3.72t
y'(1) = 3.28
To find the average velocity over each given time interval, we need to calculate the change in height and divide it by the change in time.
(a)
(i) [1, 2]:
To find the average velocity over this interval, we need to calculate the change in height and divide it by the change in time.
Change in height = y(2) - y(1) = (7(2) - 1.86(2^2)) - (7(1) - 1.86(1^2)) = 12.28 - 5.14 = 7.14 meters.
Change in time = 2 - 1 = 1 second.
Average velocity = (change in height) / (change in time) = 7.14 / 1 = 7.14 m/s.
(ii) [1, 1.5]:
Change in height = y(1.5) - y(1) = (7(1.5) - 1.86(1.5^2)) - (7(1) - 1.86(1^2)) = 8.775 - 5.14 = 3.635 meters.
Change in time = 1.5 - 1 = 0.5 seconds.
Average velocity = (change in height) / (change in time) = 3.635 / 0.5 = 7.27 m/s.
(iii) [1, 1.1]:
Change in height = y(1.1) - y(1) = (7(1.1) - 1.86(1.1^2)) - (7(1) - 1.86(1^2)) = 6.43 - 5.14 = 1.29 meters.
Change in time = 1.1 - 1 = 0.1 second.
Average velocity = (change in height) / (change in time) = 1.29 / 0.1 = 12.9 m/s.
(iv) [1, 1.01]:
Change in height = y(1.01) - y(1) = (7(1.01) - 1.86(1.01^2)) - (7(1) - 1.86(1^2)) = 6.1346 - 5.14 = 0.9946 meters.
Change in time = 1.01 - 1 = 0.01 second.
Average velocity = (change in height) / (change in time) = 0.9946 / 0.01 = 99.46 m/s.
(v) [1, 1.001]:
Change in height = y(1.001) - y(1) = (7(1.001) - 1.86(1.001^2)) - (7(1) - 1.86(1^2)) = 6.013211 - 5.14 = 0.873211 meters.
Change in time = 1.001 - 1 = 0.001 second.
Average velocity = (change in height) / (change in time) = 0.873211 / 0.001 = 873.211 m/s.
(b) To estimate the instantaneous velocity when t = 1, we can find the average velocity over a very small time interval centered around t = 1. Let's use a time interval of [1, 1.0001].
Change in height = y(1.0001) - y(1) = (7(1.0001) - 1.86(1.0001^2)) - (7(1) - 1.86(1^2)) = 6.007114 - 5.14 = 0.867114 meters.
Change in time = 1.0001 - 1 = 0.0001 second.
Average velocity = (change in height) / (change in time) = 0.867114 / 0.0001 = 8671.14 m/s.
Therefore, the estimated instantaneous velocity when t = 1 is 8671.14 m/s.
To find the average velocity over a given time interval, you need to calculate the change in position (in this case, height) and divide it by the change in time.
(a)
(i) [1, 2]:
To calculate the average velocity between times t = 1 and t = 2, we first need to find the position at these two times using the given equation y = 7t - 1.86t^2.
At t = 1: y = 7(1) - 1.86(1)^2 = 5.14 meters
At t = 2: y = 7(2) - 1.86(2)^2 = 8.24 meters
The change in position (Δy) is 8.24 - 5.14 = 3.1 meters. The change in time (Δt) is 2 - 1 = 1 second.
Average velocity = Δy / Δt = 3.1 / 1 = 3.1 m/s (rounded to two decimal places).
(ii) [1, 1.5]:
We'll follow the same process to calculate the average velocity between t = 1 and t = 1.5.
At t = 1: y = 7(1) - 1.86(1)^2 = 5.14 meters
At t = 1.5: y = 7(1.5) - 1.86(1.5)^2 = 6.675 meters
The change in position (Δy) is 6.675 - 5.14 = 1.535 meters. The change in time (Δt) is 1.5 - 1 = 0.5 seconds.
Average velocity = Δy / Δt = 1.535 / 0.5 = 3.07 m/s (rounded to two decimal places).
(iii) [1, 1.1]:
Using the same method, we can find the average velocity between t = 1 and t = 1.1.
At t = 1: y = 7(1) - 1.86(1)^2 = 5.14 meters
At t = 1.1: y = 7(1.1) - 1.86(1.1)^2 = 5.8294 meters
The change in position (Δy) is 5.8294 - 5.14 = 0.6894 meters. The change in time (Δt) is 1.1 - 1 = 0.1 seconds.
Average velocity = Δy / Δt = 0.6894 / 0.1 = 6.89 m/s (rounded to two decimal places).
(iv) [1, 1.01]:
Following the same steps as above:
At t = 1: y = 7(1) - 1.86(1)^2 = 5.14 meters
At t = 1.01: y = 7(1.01) - 1.86(1.01)^2 = 5.1694 meters
The change in position (Δy) is 5.1694 - 5.14 = 0.0294 meters. The change in time (Δt) is 1.01 - 1 = 0.01 seconds.
Average velocity = Δy / Δt = 0.0294 / 0.01 = 2.94 m/s (rounded to two decimal places).
(v) [1, 1.001]:
Using the same method:
At t = 1: y = 7(1) - 1.86(1)^2 = 5.14 meters
At t = 1.001: y = 7(1.001) - 1.86(1.001)^2 = 5.14334 meters
The change in position (Δy) is 5.14334 - 5.14 = 0.00334 meters. The change in time (Δt) is 1.001 - 1 = 0.001 seconds.
Average velocity = Δy / Δt = 0.00334 / 0.001 = 3.34 m/s (rounded to two decimal places).
(b)
To estimate the instantaneous velocity when t = 1, we can calculate the average velocity over a very small time interval centered around t = 1. For example, we can use the average velocity between [0.9, 1.1].
At t = 0.9: y = 7(0.9) - 1.86(0.9)^2 = 5.0114 meters
At t = 1.1: y = 7(1.1) - 1.86(1.1)^2 = 5.8294 meters
The change in position (Δy) is 5.8294 - 5.0114 = 0.818 meters. The change in time (Δt) is 1.1 - 0.9 = 0.2 seconds.
Average velocity = Δy / Δt = 0.818 / 0.2 = 4.09 m/s (rounded to two decimal places).
This provides an estimate for the instantaneous velocity when t = 1.