I\u2019m Andrew Hogan. These articles begin with ordinary things: a bouncing ball, a drive, a triangle, the steps of long division. Each offers a question to try and some thoughts about where it might lead.<\/p>\n
You can work through an exercise, follow a construction, or simply read. Bring your experience. There may be something here that looks different from where you are standing.<\/p>\n
A ball bounces on a piece of stretched balloon rubber covering a glass. It rises, falls, and rises again.<\/p>\n
Watch the height of each bounce. What changes? Where might the energy go? What would have to be different for the bouncing to continue forever?<\/p>\n
A small experiment can give us something concrete to observe and something much larger to imagine.<\/p>\n
Read Balloon Over Glass \u2192<\/a><\/p>\n At highway speed, a car travels a considerable distance while its driver notices a hazard and responds. Braking takes more distance still.<\/p>\n But stopping before a stationary obstacle and following another moving car are different situations. What assumptions connect them\u2014and where do those assumptions stop helping?<\/p>\n This article combines a simplified calculation, a personal experience, and time-based following guidance.<\/p>\n Read Car Lengths \u2192<\/a><\/p>\n Your One Billionth Birth-Second<\/strong><\/a> Where Do You Draw the Line?<\/strong><\/a> The Forbidden Triangle<\/strong><\/a> Robert\u2019s Remainder<\/strong><\/a> Car Lengths<\/strong><\/a> Tight Turn Limit<\/strong><\/a> Balloon Over Glass<\/strong><\/a> Charged Sphere Pump<\/strong><\/a> Most articles follow a familiar path: an exercise, an explanation, and some thoughts or further questions.<\/p>\n Before looking at an answer, you might make a prediction. While reading, notice which conditions are given and which are assumed. Afterward, try changing one condition. Does the conclusion still follow?<\/p>\n A model gives us a way to examine part of the world. Its assumptions help us understand where its conclusions apply. In the driving and cycling articles, calculated limits are learning tools; practical decisions also require attention to actual conditions and current safety guidance.<\/p>\n Sometimes a useful result is an answer. Sometimes it is a clearer question, a corrected calculation, or a new thing to make.<\/p>\n These pages grew from teaching, conversations, personal experiences, and the contributions of students. I try to give those contributions room to travel.<\/p>\n If you find an error, a missing assumption, a different construction, or a question worth adding, I would like to hear about it. Pointing to a particular sentence or step helps me understand what you noticed.<\/p>\n Email me at mr@ahogan.org<\/a> \u00b7 About this site<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" I\u2019m Andrew Hogan. These articles begin with ordinary things: a bouncing ball, a drive, a triangle, the steps of long division. Each offers a question to try and some thoughts about where it might lead. You can work through an exercise, follow a construction, or simply read. Bring your experience. There may be something here […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-203","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/pages\/203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/comments?post=203"}],"version-history":[{"count":6,"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/pages\/203\/revisions"}],"predecessor-version":[{"id":229,"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/pages\/203\/revisions\/229"}],"wp:attachment":[{"href":"https:\/\/ahogan.org\/index.php\/wp-json\/wp\/v2\/media?parent=203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}A question with everyday consequences<\/h2>\n
How much room does it take to stop?<\/h3>\n
Explore the articles<\/h2>\n
Time and the numbers we live with<\/h3>\n
\nWe usually count our lives in years. What becomes noticeable when we count in seconds?<\/p>\n
\nA drive, a traffic report, and a few minutes saved. What exactly is being measured when we talk about time?<\/p>\nMaking something from a possibility<\/h3>\n
\nTwo sides, a missing length, and a tent made from unequal halves. What might we build when more than one answer is possible?<\/p>\n
\nLong division leaves a trail of intermediate results. What happens when we collect them into a new number?<\/p>\nMotion, limits, and physical models<\/h3>\n
\nExplore the distance traveled before and during braking, and why following distance is measured usefully in seconds.<\/p>\n
\nA bicycle\u2019s speed, the radius of a turn, and the grip between tires and road. How can a calculation help us think about a physical limit?<\/p>\n
\nA bouncing ball opens questions about energy, observation, and the possibility of forever.<\/p>\n
\nTouching two spheres redistributes their charge. What can a sequence of these simple operations accomplish?<\/p>\nA way to explore<\/h2>\n
Keep the conversation going<\/h2>\n