If a rock is thrown upward on the planet Mercury with a velocity of 11 m/s, its height in meters t seconds later is given by y = 11t - 1.86t sqaure
a : Find the average velocity over the given time intervals: [1, 2] , [1, 1.5] , [1, 1.1] , [1, 1.01] [1, 1.001]
b: Estimate the instantaneous velocity when t = 1.
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To find the average velocity over the given time intervals, we need to find the average change in height over each interval.
a) Average velocity over the interval [1, 2]:
We can calculate the height at t = 1 and t = 2 and find the difference:
y(1) = 11(1) - 1.86(1)^2 = 9.14 m
y(2) = 11(2) - 1.86(2)^2 = 14.24 m
Average velocity = (y(2) - y(1)) / (2 - 1) = (14.24 - 9.14) / (2 - 1) = 5.1 m/s
b) Average velocity over the interval [1, 1.5]:
We can calculate the height at t = 1 and t = 1.5 and find the difference:
y(1) = 9.14 m (this value was already calculated)
y(1.5) = 11(1.5) - 1.86(1.5)^2 = 11.025 m
Average velocity = (y(1.5) - y(1)) / (1.5 - 1) = (11.025 - 9.14) / (1.5 - 1) = 4.885 m/s
c) Average velocity over the interval [1, 1.1]:
We can calculate the height at t = 1 and t = 1.1 and find the difference:
y(1) = 9.14 m (this value was already calculated)
y(1.1) = 11(1.1) - 1.86(1.1)^2 = 9.08 m
Average velocity = (y(1.1) - y(1)) / (1.1 - 1) = (9.08 - 9.14) / (1.1 - 1) = -0.06 m/s
d) Average velocity over the interval [1, 1.01]:
We can calculate the height at t = 1 and t = 1.01 and find the difference:
y(1) = 9.14 m (this value was already calculated)
y(1.01) = 11(1.01) - 1.86(1.01)^2 = 9.1536 m
Average velocity = (y(1.01) - y(1)) / (1.01 - 1) = (9.1536 - 9.14) / (1.01 - 1) = 0.147 m/s
e) Average velocity over the interval [1, 1.001]:
We can calculate the height at t = 1 and t = 1.001 and find the difference:
y(1) = 9.14 m (this value was already calculated)
y(1.001) = 11(1.001) - 1.86(1.001)^2 = 9.14414 m
Average velocity = (y(1.001) - y(1)) / (1.001 - 1) = (9.14414 - 9.14) / (1.001 - 1) = 0.414 m/s
b) To estimate the instantaneous velocity when t = 1, we can calculate the velocity at that point by finding the derivative of the height function.
Given the equation: y = 11t - 1.86t^2
Taking the derivative with respect to t:
dy/dt = 11 - 2(1.86t)
Substituting t = 1 into the derivative:
dy/dt | t=1 = 11 - 2(1.86)(1) = 11 - 3.72 = 7.28 m/s
So, the estimated instantaneous velocity when t = 1 is 7.28 m/s.
a) To find the average velocity over a given time interval, you need to calculate the change in position (height) divided by the change in time.
For the time interval [1, 2]:
The initial time is t = 1, and the final time is t = 2.
To find the change in height over this interval, substitute the initial and final times into the equation:
y(2) - y(1) = (11(2) - 1.86(2)^2) - (11(1) - 1.86(1)^2)
Simplify the equation:
(22 - 7.44) - (11 - 1.86) = 14.56 - 9.14 = 5.42 meters
The change in height over this interval is 5.42 meters.
Now, calculate the change in time:
Δt = 2 - 1 = 1 second
Finally, divide the change in position by the change in time to find the average velocity:
Average velocity = (change in position) / (change in time) = 5.42 meters / 1 second = 5.42 m/s
For the remaining time intervals ([1, 1.5], [1, 1.1], [1, 1.01], [1, 1.001]), repeat the same process of substituting the initial and final times into the equation, calculating the change in height, change in time, and then finding the average velocity.
For the time interval [1, 1.5]:
Substitute t = 1 and t = 1.5 into the equation, and calculate the change in height. Then calculate the change in time. Finally, divide the change in position by the change in time to find the average velocity.
For the remaining time intervals [1, 1.1], [1, 1.01], and [1, 1.001], repeat the same process as described above.
b) To estimate the instantaneous velocity when t = 1, you can use the concept of the derivative. The instantaneous velocity is given by the derivative of the position function with respect to time.
The position function in this case is y = 11t - 1.86t^2.
To find the derivative of this function, separate each term and apply the power rule:
dy/dt = 11 * d/dt(t) - 1.86 * d/dt(t^2)
= 11 - 1.86 * 2t
= 11 - 3.72t
Substitute t = 1 into the derivative to find the instantaneous velocity when t = 1:
Velocity at t = 1 = 11 - 3.72(1)
= 11 - 3.72
= 7.28 m/s
Therefore, the estimated instantaneous velocity when t = 1 is 7.28 m/s.