{"id":1778,"date":"2015-04-22T12:17:44","date_gmt":"2015-04-22T18:17:44","guid":{"rendered":"http:\/\/www.ghyzmo.com\/?p=1778"},"modified":"2015-05-02T12:50:17","modified_gmt":"2015-05-02T18:50:17","slug":"an-algebraic-discussion-of-work-and-energy","status":"publish","type":"post","link":"http:\/\/www.ghyzmo.com\/an-algebraic-discussion-of-work-and-energy\/","title":{"rendered":"An Algebraic discussion of Work and Energy"},"content":{"rendered":"

\u00a9Fernando Caracena<\/em> 2015<\/p>\n

This discussion parallels the post on \"Work and Energy<\/a>\", but uses algebra instead of calculus. The discussion is simplified by making simplifying assumptions to arrive at the same conclusions. The discussion involving calculus is more general and not so specialized. Simplicity here is gained by a loss of generality.<\/p>\n

Preliminaries<\/strong><\/em><\/h2>\n

This discussion builds onto that contained in a previous post\u00a0 Grokking Galileo's Physics part II.<\/a> It does so in a way that Galileo could have understood because he did not know calculus, which was developed by Newton and Leibnitz.<\/a> However, Galileo was a pretty smart guy, and would have been able to learn calculus fast, in which case he would have probably preferred the discussion involving calculus. Here we use a very simplified model for our\u00a0 discussion.<\/p>\n

Limiting Assumptions<\/strong><\/em><\/h3>\n

Consider a mass m, which is moving to the right along a straight line, the x-axis, starting from x=0 at t=0 with an initial speed of\u00a0 uo<\/sub>. A constant force F acts to the right along this line, which produces an acceleration a.<\/p>\n

 <\/p>\n

Derivation<\/strong><\/em><\/h2>\n

At a constant acceleration, a, and at any time, t > 0, the mass has a speed as a function of time,<\/p>\n

u =\u00a0uo<\/sub> + a t. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (1)<\/p>\n

The average speed of the mass during the time, t, is<\/p>\n

<u>=\u00bd (u + uo<\/sub>). \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2)<\/p>\n

The distance travelled from the origin (the position at t=0) is just<\/p>\n

x= <u> t.\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3a)<\/p>\n

Substituting (2) in (3a), we have<\/p>\n

x= \u00bd (u + uo<\/sub>) t ,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3b)<\/p>\n

which with the further substitution of (1) reduces to<\/p>\n

x = uo<\/sub> t + \u00bd a t<\/strong>2<\/sup><\/strong><\/strong>. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (3c)<\/p>\n

The above (3c) is just the equation for the position of a mass moving under constant acceleration to the right along the x-axis after starting from the origin, with an initial velocity of uo<\/sub>.<\/p>\n

It is possible to eliminate time from the description of this motion by solving for it using (1),<\/p>\n

t = (u - uo<\/sub> )\/a , \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (4)<\/p>\n

and plugging that result and (2) back into (3),<\/p>\n

x= \u00bd (u + uo<\/sub>) (u - uo<\/sub> )\/a . \u00a0\u00a0 (5a)<\/p>\n

carrying through the multiplication and solving for u in terms of x, we get the following:<\/p>\n

u2<\/sup><\/strong><\/strong> = \u00a0 uo<\/sub>2<\/sup><\/strong><\/strong> + 2 a x. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (5b)<\/p>\n

At this point, multiplying both sides of (5b) by\u00a0 \u00bd m gives the results,<\/p>\n

\u00bd m u2<\/sup><\/strong><\/strong> = \u00bd m uo<\/sub>2<\/sup><\/strong><\/strong> + (m a) x. \u00a0 (6a)<\/p>\n

Identifying the term in parenthesis on the right hand side (RHS) of (6a) as the force, from Newton's Laws of motion, results in,<\/p>\n

\u00bd m u2<\/sup><\/strong><\/strong> = \u00bd m uo<\/sub>2<\/sup><\/strong><\/strong> +F x.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6b)<\/p>\n

By defining the work done by the constant force F as,<\/p>\n

W= F x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6c)<\/p>\n

and kinetic energy as,<\/p>\n

KE=\u00bd m u2<\/sup><\/strong><\/strong>, \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6d)<\/p>\n

we can further simplify (6b) to<\/p>\n

KE =\u00a0 KEo<\/sub> +W. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (7)<\/p>\n

Note that (7) identifies work as the transfer of energy through the agency of an external force acting to change the state of motion of a mass.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u00a9Fernando Caracena 2015 This discussion parallels the post on \"Work and Energy\", but uses algebra instead of calculus. The discussion is simplified by making simplifying assumptions to arrive at the same conclusions. The discussion involving calculus is more general and … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,60,10,12],"tags":[],"class_list":["post-1778","post","type-post","status-publish","format-standard","hentry","category-algebra","category-energy","category-mathematics","category-physics"],"_links":{"self":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts\/1778","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/comments?post=1778"}],"version-history":[{"count":4,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts\/1778\/revisions"}],"predecessor-version":[{"id":1782,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts\/1778\/revisions\/1782"}],"wp:attachment":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/media?parent=1778"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/categories?post=1778"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/tags?post=1778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}