A ball is thrown upward at time t = 9 seconds on planet Ubuntu, and its acceleration from time t=9 seconds to t=21 seconds is given by the function .
a(t) = (3) / (t - 8) m/s^2
Use 4 time intervals of equal length to overestimate, and then underestimate, the relative velocity of the rising ball from time t = 9 sec to t = 21 seconds.
(Note: ``relative'' means relative to the initial velocity at t = 9, which has not been given by the problem.)
a. Overestimate = ? m/sec.
b. underestimate = ? m/sec.
a nice calculator for this is at
http://mathworld.wolfram.com/RiemannSum.html
To find the overestimate and underestimate of the relative velocity of the rising ball from time t = 9 sec to t = 21 seconds, we need to calculate the change in velocity over equal time intervals.
First, let's calculate the change in velocity for each time interval.
We have the acceleration function:
a(t) = 3 / (t - 8) m/s^2
Now, we need to integrate the acceleration function to get the velocity function:
v(t) = ∫a(t) dt
Integrating 3 / (t - 8) dt gives:
v(t) = 3 ln|t - 8| + C
Since we are considering the relative velocity of the rising ball from time t = 9 sec to t = 21 seconds, we need to find the value of the constant C.
To find C, we can use the initial condition that the velocity at t = 9 seconds is equal to the initial velocity of the ball:
v(9) = C
The initial velocity of the ball is not given in the problem statement, so we don't have enough information to calculate its value.
However, we can still proceed by considering only the change in velocity for each time interval, without knowing the initial velocity.
Let's calculate the change in velocity for each interval:
Interval 1: t = 9 sec to t = 12 sec
Δv1 = v(12) - v(9)
Interval 2: t = 12 sec to t = 15 sec
Δv2 = v(15) - v(12)
Interval 3: t = 15 sec to t = 18 sec
Δv3 = v(18) - v(15)
Interval 4: t = 18 sec to t = 21 sec
Δv4 = v(21) - v(18)
Now, let's calculate these changes in velocity using the velocity function we obtained earlier.
Δv1 = (3 ln|12 - 8| + C) - (3 ln|9 - 8| + C)
Simplifying:
Δv1 = 3 ln(4) - 3 ln(1)
Δv1 = 3 ln(4)
Similarly, we can calculate Δv2, Δv3, and Δv4 using the same process.
Now, to overestimate the relative velocity, we can sum up the changes in velocity:
Overestimate = Δv1 + Δv2 + Δv3 + Δv4
To underestimate the relative velocity, we can take the absolute value of each change in velocity and sum them:
Underestimate = |Δv1| + |Δv2| + |Δv3| + |Δv4|
Unfortunately, without the initial velocity of the ball, we cannot calculate the exact values of the overestimate and underestimate. We can only provide the general calculation process described above.