Trajectory Equation and Envelope Equation

[@Infinium]

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[@gibbogle]

The figure showing trajectories is very misleading, because it shows the trajectory with the lowest elevation (smallest alpha) going as far as the trajectory above it. The maximum distance is achieved at some alpha (I think it's alpha = pi/4), and for every alpha less than that the distance decreases as alpha decreases - these trajectories do not touch the envelope.

[@Infinium]

Hi! Yes, I am aware of this, I realised after publication of this video, sorry!

[@joeeeee8738]

Great video!! Simple ideas greatly explained one after the other

[@Infinium]

Thank you ❤️

[@iiiiii-w8h]

Why is Paul Bettany narrating this? Can you say "What is beef if not cow persevering?"

[@Infinium]

I’m not actually aware of who this is haha! Maybe in the future ;)

[@MathPhysicsEngineering]

I made a video exactly on the same subject. Check the playlist on my channel called projectile motion in depth.

[@Infinium]

Oh amazing, I’ll go check it out!

[@anshugupta793]

Sir i didn't understand why does it sits within the boundary in ∆>0 5:25 Can i give the reason that since Height max=V²sin²x/(2g) =z Noticing that for some constant z we get a unique x which satisfies hence ∆=0 for height max

[@logician1234]

Is this an equation of an ellipse?

[@Infinium]

It’s not necessarily - in the video, it looks like an ellipse for simplicity!

[@srevere7241]

Actually, if you find the equation that covers all max heights of all the trajectories, you get a perfect ellipse.

[@cowgomoo444]

wow! great vid. z(x) seems to be of the form ax^2+b. this means its a parabola right? V is not a variable here its a parameter. anyways really great vid. i was reading about envelopes in my math textbook and i can confidently say that this vid took me from 85% understanding to 99% understanding. thanks

[@Infinium]

Thank you for your comment, you are correct! I'm glad this video could help improve your understanding! :)

[@cowgomoo444]

@@Infinium no worries. you earned yourself a subscriber

[@Infinium]

Thank you ❤️

[@joeeeee8738]

What's the formal definition of an envelope?

[@Mugilans-sq4uk]

thanks a lot sir

[@Infinium]

No problem! :)

[@Mugilans-sq4uk]

😃😃

[@Pedritox0953]

Very nice video

[@Infinium]

Thank you ❤️

[@sythatsokmontrey8879]

Very great video.

[@Infinium]

Thanks soo much!

[@TheJara123]

Brilliant

[@Infinium]

Thank you ❤️

[@lgent2435]

Great ideas and logics

[@Infinium]

Thank you ❤️