Applications of Slope Fields (Differential Equations 10)

How Slope Fields can be applied to Real-Life situations and model solutions and long-term trends.

[sriramn]

Binge watched the first 10 videos at 2x with a little skipping of the routine parts. Simply superb. No need to take notes. It just makes sense. Beautiful rhythm, intonation, gestures, movement. Master class.

[ArleynH]

I'm getting closer to zen with each second of these videos

[HritikJain]

You actually explained physics better than my physics teacher.

[donedeal]

Hmmm so parachute design is all about changing the K...Thanks Prof Leonard!

[ramibanyahmad]

Thank you so much for these valuable videos, your efforts, and your time.

[novikbenjamin]

This episode, professor Superman is going to talk about how to be a Superman.

[vulkanpeter-brezovsky]

The formula for air resistance(or drag, generally) is the following: F_d=1/2*density*surface area*drag coefficient(the role of the shape, basically)*velocity^2. Velocity squared! That means that air resistance isn’t proportional to velocity, rather velocity squared. You could see at any point of the video, that the units of measurement don’t match up. That’s because what he wrote as dv/dt=g-kv isn’t true( or reflective of the real world). Anyways, the mathematical explanation is still right, so if we theoreticized a world where air resistance is proportional to velocity in a linear way, then ok. But the physics behind it is not accurate. In the previous application video, in one of the examples v=-10 as he writes it, but that doesn’t give the answer that he wrote. That’s because it’s positive 10.

[ezbeanss]

0:56 "If there were no air resistance, we would pick up velocity until we smash into the ground... or pull a parachute." Listen man I don't think there is any easy way to say it but a parachute aint gonna do anything for you there if there's no air resistance...

[mnuman]

World Teacher's Day (Professor Leonard) 5 October.

[franciscopen]

This video was excellent, because he didn’t give the full picture, you had to crunch it.. so the Y was implicit, you don’t how hot it relates to the independent variable yet, and if you paid attention to the first videos, you would know that a coefficient times rate of change, integrate that and you get an exponential, and the exponential shows here graphically, and it just makes sense.

[SaifUlIslam-dbnu]

Thank you for the lecture. It is definitely helping me out understand 'linearity' and the theorem, dy/dx + P(x)*y = f(x) . Just a question, for the first question, that is dv/dt = 32 - 1.6v, why did you not substitute v = distance/time. Since we are 'leaning' on 't' as our independent variable, why not just make the right side explicit by writing it in the form, dv/dt = 32 - 1.6*Distance/Time .

My intuition is that since we want to plot a graph, and we want to include 'm' ( slope ) as an axis, we let 'm' represent the x axis. Now since anything along the 'x' axis acts as an independent variable, we cannot let 't' be on the right side, that would mean that there are two independent variables in the same equation, each on a different side. Thus, letting 'v' be the y axis ( on the right side ) would make the most sense.

I'm on the fence about this idea, not completely sure.

Can anyone help out? Thanks!

[KD-oppb]

Hope that Laplace Transform is coming soon Prof!

[TheRandomFool]

Solving the helicopter differential equation exactly, it can be shown that in theory you never actually reach the point where your acceleration reaches 0 after jumping out of the helicopter. However, in reality, variations in the air and various currents will cause you to reach terminal velocity and continue to accelerate and decelerate slightly. Tumbling will affect this a fair bit. G and air density is assumed to be constant because of the relatively small scale. It can get pretty complicated if you try to take everything into account.

[ajaib]

Awesome video sir. You are a great soul .😎😍😍. I learned more from you. 😘😘😘

[DEATHHAWK]

Thank for the lecture, helped with my research paper due for my college course. But GOD DAMN them biceps are freaking huge!

[binodpachhai]

Its necessary to all thanks for sharing sir

[johncurran]

@1:00, if you pull a parachute with no air resistance you still go splat.

[moehassan_]

I'm quite confused on the concept that since t are wrapped up in functions of P and v, that when t increases but P and v are constant that our slopes remain the exact same. I tried making an explanation for this fact but I'm not sure if it is valid or makes intuitive sense why.

We have to remember that P and v are functions of time, and t is contained within these functions. If t is changing, P and v must change as well. Thus it makes sense that when P or v are held constant, that t cannot change either. Thus dP/dt and dv/dt are constant.

If anyone has anything to add on please do! this is part of the lesson perplexed me.

[potatolegs]

wouldn’t the slopes scale with the y-axis? so like a slope of 32 would look like approx. like a slope of 6 on this because 6 boxes represent 30 ft/s.

[tangsoopap]

I don't understand what you get with g - kv when you take an acceleration (m/sec^2) and subtract a velocity (m/sec). They're not even the same units...unless somehow k is a pure number/sec...but what does that mean?