Vector Functions: Problems Plus
Published on HivePostify by @mes · Sun Aug 30 2026
https://play.3speak.tv/embed?v=mes/f1pdtn54
In this video, I go over the end-of-chapter Problems Plus questions and solve 7 very advanced problems on vector functions. Three of the questions involve launching a projectile either on a flat, inclined, or declined plane. One question is on curvature, one on finding the equation of a plane, another is on rolling a ball off a table, and my favorite one is on finding the axial distance covered when wrapping a cable of known length around a spool of known diameter. Buckle up for 5 hours of mathematics!
[3Speak](https://3speak.tv/watch?v=mes/vector-functions-problems-p-729) - [YouTube](https://youtu.be/YJRehWJxcEY) - [Telegram](https://t.me/meslinks/35445) - [Odysee](https://odysee.com/@mes:8/Vector-Functions-Problems-Plus:6) - [BitChute](https://www.bitchute.com/video/LCcn2MoQW3ti/) - [Rumble](https://rumble.com/v7evawc-vector-functions-problems-plus.html) - [Notes](https://mes.fm/vector-functions-problems-plus) ([PDF](https://1drv.ms/b/c/88862ef47bcaf6cd/IQBeffZ0TE-QQ6lmLsyB-FCDAQSzCsLke1J1S4HQcnoIAH4)) - [Problems Plus](https://www.youtube.com/playlist?list=PLHNdnBXVWSlo) - [Vector Functions](https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBSO8Te8PcpNm4U) - [mes.fm/math](https://mes.fm/math)
#math #calculus #vectorfunctions #problemsplus #education
Timestamps
- Intro – 0:00 - Calculus book reference – 0:35 - Calculus book chapter – 1:03 - Topics to cover – 2:15 - Problems Plus: 7 Problems – 2:50 - Problem 1: Projectile fired from origin – 12:04 - Problem 2: Projectile fired down inclined plane – 1:26:47 - Problem 3: Ball rolls of table – 2:26:37 - Problem 4: Curvature from parametric equations – 3:02:35 - Problem 5: Total distance traveled by projectile – 3:15:35 - Problem 6: Cable wound around a spool – 4:17:17 - Problem 7: Vector equation on a plane – 5:02:49 - Outro: 5:17:47
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Links and Calculus Book Chapter
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>- MES Math Tutorials: [mes.fm/math](https://mes.fm/math) >- [MES Links](https://mes.fm/links) - [Donate](https://mes.fm/donate) - [Telegram](https://t.me/meslinks) - @mestruth on [Hive](https://3speak.tv/user/mestruth) & [YouTube](https://www.youtube.com/@mestruth) - [Hive blockchain tutorial](https://peakd.com/hive-181335/@mes/tdvxnmej) - [Video playback speed tutorial](https://peakd.com/video/@mes/play-videos-at-faster-or-slower-speeds-on-any-website) - [Where Did the Towers Go? by Dr. Judy Wood](http://wheredidthetowersgo.com) + MANY more links… >- Reuse of my videos: Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video. >- Fight back against censorship: Bookmark sites/channels/accounts and check periodically. - [Archive webpages](https://chrome.google.com/webstore/detail/archive-page/gcaimhkfmliahedmeklebabdgagipbia) - [ArcHive](https://dhenz14.github.io/archive/index.html)
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Calculus Book Reference
Note that I mainly follow along the book:
- Calculus: Early Transcendentals 7th Edition by James Stewart: [Link](https://web.archive.org/web/20250216211115/https://patemath.weebly.com/uploads/5/2/5/8/52589185/james-stewart-calculus-early-transcendentals-7th-edition-2012-1-20ng7to-1ck11on.pdf) - I used the following solution manual for this chapter: [Link](https://web.ma.utexas.edu/users/shirley/a408d/hwlist/1hw/Addendum/Stewart's%20Calculus,%209th%20ed,%20Soln's%20Manuals/Solution%20Manual,%20ch's%2010-16/StewartCalcET9eSAGCH13.pdf) - I used Quizlet.com 30 day free trial for the solutions to the Concept Check: [Link](https://theswissbay.ch/pdf/Gentoomen%20Library/Maths/Calculus/Calculus%20-%20Early%20Transcendentals%206e.pdf) - Note: In some earlier videos I used the [6th edition](https://theswissbay.ch/pdf/Gentoomen%20Library/Maths/Calculus/Calculus%20-%20Early%20Transcendentals%206e.pdf).
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Calculus Book Chapter
The Hive notes and sections playlist for each video of this chapter are listed below:
[Vector Functions ▶️](https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBSO8Te8PcpNm4U)
1. [Vector Functions and Space Curves](https://peakd.com/hive-128780/@mes/vector-functions-space-curves) - [▶️](https://www.youtube.com/playlist?list=PLai3U8-WIK0GwdMY-E94LsmYmO-IxcoX) 2. [Derivatives and Integrals of Vector Functions](https://peakd.com/hive-128780/@mes/hfsetrue) - [▶️](https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous) 3. [Arc Length and Curvature](https://peakd.com/hive-128780/@mes/koxharrd) - [▶️](https://www.youtube.com/playlist?list=PLai3U8-WIK0Gd5OieYiZMaSKw1QfyHbiG) 4. [Motion in Space: Velocity and Acceleration](https://peakd.com/hive-128780/@mes/e443f9d3) - [▶️](https://www.youtube.com/playlist?list=PLai3U8-WIK0Hxo8aPU3ezG0tcLvgJnxLB) - [Applied Project: Kepler's Laws](https://peakd.com/hive-128780/@mes/c3b94a80) - [▶️](https://www.youtube.com/playlist?list=PLai3U8-WIK0FSh63mgXOKLQ41H4dkzbaY) 5. [Review](https://peakd.com/hive-128780/@mes/review-of-vector-functions-138) - [▶️](https://www.youtube.com/playlist?list=PLAsz0y2lc-c4) — - Concept Check - True-False Quiz 6. [Problems Plus](https://mes.fm/vector-functions-problems-plus) - [▶️](https://www.youtube.com/playlist?list=PLHNdnBXVWSlo) – CURRENT VIDEO
This links are also on MES Links: https://mes.fm/links
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Topics to Cover
- Problem 1: Projectile fired from origin - Problem 2: Projectile fired down inclined plane - Problem 3: Ball rolls off table - Problem 4: Curvature from Parametric Equations - Problem 5: Total distance traveled by projectile - Problem 6: Cable wound around a spool - Problem 7: Vector equation on a plane
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Problems Plus
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Problem 1: Projectile fired from origin
A projectile is fired from the origin with angle of elevation α and initial speed v 0 .
Assuming that air resistance is negligible and that the only force acting on the projectile is gravity, g, we showed in my [earlier video](https://peakd.com/hive-128780/@mes/e443f9d3) that the position vector of the projectile is:
We also showed that the maximum horizontal distance (or range) of the projectile is achieved when α = 45° and in this case the range is R = (v 0 ) 2 / g.
(a) At what angle should the projectile be fired to achieve maximum height and what is the maximum height?
(b) Fix the initial speed v0 and consider the parabola x 2 + 2Ry - R 2 = 0, whose graph is shown in the figure below.
Show that the projectile can hit any target inside or on the boundary of the region bounded by the parabola and the x-axis, and that it can't hit any target outside this region.
(c) Suppose that the gun is elevated at an angle of inclination α in order to aim at a target that is suspended at a height h directly over a point D units downrange.
The target is released at the instant the gun is fired.
Show that the projectile always hits the target, regardless of the value v0, provided the projectile does not hit the ground "before" D.
Problem 2: Projectile fired down inclined plane
(a) A projectile is fired from the origin down an inclined plane that makes an angle θ with the horizontal.
The angle of elevation of the gun and initial speed of the projectile are α and v0, respectively.
Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of time t.
>Ignore air resistance.
(b) Show that the angle of elevation α that will maximize the downhill range is the angle halfway between the plane and the vertical.
(c) Suppose the projectile is fired up an inclined plane whose angle of inclination is θ.
Show that, in order to maximize the (uphill) range, the projectile should be fired in the direction halfway between the plane and the vertical.
(d) In a paper presented in 1686, Edmond Halley summarized the laws of gravity and projectile motion and applied them to gunnery.
One problem he posed involved firing a projectile to hit a target a distance R up an inclined plane.
Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angel in part (c).
>Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.
Problem 3: Ball rolls off table
A ball rolls off a table with a speed of 2 ft/s.
Tags: #math#calculus#science#education#vectorfunctions#vectors#problemsplus#scifimultiverse#chessbrothers#hive-128780