Free fall with air resistance An object is dropped straight...

Free fall with air resistance An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in $\mathrm{ft} / \sec ^{2}$ and recorded every second after the drop for $5 \mathrm{sec},$ as shown: $$ \begin{array}{l}{\text { a. Find an upper estimate for the speed when } t=5 \text { . }} \\ {\text { b. Find a lower estimate for the speed when } t=5 \text { . }} \\ {\text { c. Find an upper estimate for the distance fallen when } t=3 \text { . }}\end{array} $$

Free fall with air resistance An object is dropped straight down
from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of
air resistance. The acceleration is measured in $\mathrm{ft} / \sec ^{2}$ and
recorded every second after the drop for $5 \mathrm{sec},$ as shown:
$$
\begin{array}{l}{\text { a. Find an upper estimate for the speed when } t=5 \text { . }} \\ {\text { b. Find a lower estimate for the speed when } t=5 \text { . }} \\ {\text { c. Find an upper estimate for the distance fallen when } t=3 \text { . }}\end{array}
$$

Key Concepts

Riemann Sums and Numerical Integration

Numerical integration using Riemann sums is a method for approximating the value of a definite integral by summing the areas of a series of rectangles under a curve. In problems where measurements or computations are only available at discrete points, such as periodic acceleration readings, Riemann sums provide a systematic way to estimate the cumulative effect, like the change in velocity or total distance traveled, by taking the sum of values multiplied by the time interval. This includes both upper and lower estimates corresponding to over- or under-estimations of the actual integral.

Relationship between Acceleration, Velocity, and Displacement

In the study of motion, acceleration, velocity, and displacement are related through the process of differentiation and integration. Acceleration is the derivative of velocity, and velocity is the derivative of displacement. Conversely, integrating the acceleration with respect to time yields the velocity, while integrating the velocity provides the displacement. These fundamental relationships allow one to compute one quantity from another, particularly in cases where acceleration is known as a function of time.

Motion Under Gravity with Air Resistance

Motion under gravity with air resistance involves a non-constant acceleration because the resistive force of air opposes the gravitational force. Unlike free fall in a vacuum, where acceleration remains constant, this scenario demonstrates that the rate of acceleration decreases over time due to the increasing influence of air resistance. This principle is crucial in understanding real-world object motion and impacts the techniques used for approximating speed and distance via integration since the changes in acceleration must be accurately accounted for over each time interval.

Transcript

00:01 All right, the question here describes an object in free fall with air resistance.

00:06 It's dropped from a helicopter.

00:08 The object continues to fall faster and faster due to the acceleration of gravity and the rate of change in its velocity decreases over time because of air resistance.

00:22 The acceleration is measured in feet per second squared and recorded every second after the drop for five seconds.

00:32 As shown.

00:33 And we have this table right here.

00:38 Now for part a, we are asked to determine the upper estimate for speed when time equals five.

00:53 So our upper limit points are going to be the ones from t1 to t5.

01:06 Or, i'm sorry, no, they'll be from t0 to t4, and those will be our five upper hundred points.

01:16 And basically we're going to calculate speed by saying, well, speed is roughly equal to the change in time multiplied by the acceleration due to time one, time initial time, plus acceleration due to time one, time two, time three, and time four.

02:07 Okay, these are the points that we showed right here we're going to use for our upper estimate and when we take the change in time and multiply it by the acceleration due to time one sorry, time zero to time four we can see we can calculate our velocity okay, and our speed so we're going to say well the change and time is one second we see it changes one second and our acceleration due to time initially it's going to be gravity which is 32 and then our air resistance slows it down so now we have 19 .41 and then it's continuously continues to slow down so finally our last point here is 4 .3 okay when we plug all this in we're going to get roughly equal to to 74 .65 feet for second as our upper estimate for speed.

03:42 Now, our lower estimate for speed will not take into account the initial time.

03:53 Instead of taking into account the initial time here, we will, however, take into account time one, and we're going to go all the way to time.

04:06 Okay, that'll be our lower estimate.

04:10 We say, well, all right, the change in time multiplied by the acceleration due to time 1 plus the acceleration due to times 2 plus acceleration due to time 3 and acceleration due to time 4.

04:32 And finally, acceleration due to time.

04:38 And we're going to get our final answer to be a, sorry, 1, as our change.

04:43 Change in time multiplied by 19 .41 plus 11 .77 plus 7 .14 plus 7 .14, 4 .33.

05:10 And then finally, we have 2 .6.

05:22 And that final answer will come out 45 .28 feet.

05:34 That's hour.

05:37 Speed.

05:40 Okay, so we can estimate the speed after five seconds to be anywhere from 45 .28 to 74 .65.

05:54 This is our lower estimate...