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 $\mathrm{m} / \mathrm{s}^{2}$ and recorded every second after the drop for $5 \mathrm{~s}$, as shown: a. Find an upper estimate for the speed when $t=5$. b. Find a lower estimate for the speed when $t=5$. c. Find an upper estimate for the distance fallen when $t=3$.

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 $\mathrm{m} / \mathrm{s}^{2}$ and recorded every second after the drop for $5 \mathrm{~s}$, as shown:
a. Find an upper estimate for the speed when $t=5$.
b. Find a lower estimate for the speed when $t=5$.
c. Find an upper estimate for the distance fallen when $t=3$.

Key Concepts

Differential Calculus and Integration

In kinematics, acceleration is the derivative of velocity with respect to time, and velocity is the derivative of displacement with respect to time. To estimate velocity from given acceleration data, one must integrate the acceleration over the time interval. Likewise, integrating velocity provides the displacement. This concept is fundamental to translating between rates of change and accumulated quantities.

Numerical Approximation Using Riemann Sums

Often, when dealing with discrete data or functions that are difficult to integrate analytically, Riemann sums offer a method of approximating the area under a curve. By partitioning the time interval and summing the contributions from each subinterval, one can estimate the integral. Upper and lower estimates (using maximum or minimum values in each subinterval) provide bounds on the actual value of the integral.

Free Fall with Air Resistance

Free fall involves motion under the influence of gravity, but the addition of air resistance alters the expected behavior by reducing acceleration over time. This concept highlights how resistive forces, such as drag, modify the motion; acceleration decreases even though the object’s speed increases, emphasizing the need to consider non-ideal forces in dynamic systems.

Transcript

00:01 Okay, what we want to do is step through a problem about acceleration, velocity, and distance.

00:12 And so what we have is we have an object that is dropped, dropped from a helicopter, and is allowed to free fall.

00:35 And its corresponding time for the first five seconds with its acceleration in feet per second squared is given in this following table.

00:46 And we want to know what the upper estimate for velocity is, what the lower estimate for velocity is, and then the distance.

00:57 And so one of the things that is going to help us is to go ahead and just draw a quick graph.

01:03 So i'm just going to draw a quick graph.

01:08 This is time in seconds.

01:10 This is going to be my acceleration in feet per second squared.

01:15 I'm going to go 30, no, 10, 20, 30.

01:20 So these are given in increments of 10.

01:23 And then these will be two, three, four, five.

01:28 So these will be in increments of one.

01:30 And at zero, i'm up here at 32.

01:34 And it doesn't have to be perfect.

01:36 One, i'm down here kind of at 19, 2, 11, 3, 7, 4, i'm down here at 4 and then 2.

01:51 So here's kind of my graph right here.

01:56 And so the first thing i want to do is to find the upper estimate for the velocity.

02:09 So we know that velocity is actually the area under the curve.

02:15 And so i want to do it for all five seconds.

02:19 And so you know that the upper estimate now is going to kind of be the left end in points.

02:25 So here would be my rectangles right here.

02:32 And this would represent my upper estimate.

02:38 And so for that purpose, my velocity is upper estimate is, and i'm doing increments, equal increments of time.

02:47 So there's my one second seconds.

02:49 And then the height of each of those rectangles is where it would be where it touches the actual graph.

02:56 So that is going to be the 32 plus the 19 plus the 11 .77 and then plus the 7 .14 and then plus the 4 .33.

03:13 And so the velocity, the upper estimate from a velocity, would be 74 .65 feet per second.

03:23 So now the second time i want to do this is i want to do the lower estimate for my velocity.

03:35 And so once again, i have equal widths, but now this is my lower estimate.

03:43 So this would be my first rectangle right here.

03:47 Then this would be my second rectangle.

03:54 Then, oh, i did this wrong.

03:56 So i need to go from this side.

03:59 So i would go from that height and then up to this height, then up to that height, then up to that height, this way, and up that way.

04:16 So here would be my green rectangles right here.

04:20 And so my first height would be here at the 263.

04:23 So i'm actually going to be starting this backwards and ending here because 19 .41 would be my last...

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