{"id":1403,"date":"2026-01-28T12:47:36","date_gmt":"2026-01-28T02:47:36","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1403"},"modified":"2026-01-28T13:31:35","modified_gmt":"2026-01-28T03:31:35","slug":"year12-math-4-4-3-modelling-motion-kinematics-dynamics","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1403","title":{"rendered":"Year12 MATH 4-4-3 modelling motion (kinematics\/dynamics)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Modelling Motion in Unit 4 of QLD Year 12 Specialist Mathematics (often Topic 4) <mark>involves using calculus and vector techniques to analyze the movement of objects, combining kinematics (description of motion) and dynamics (forces causing motion)<\/mark>. It extends previous mechanics knowledge to non-constant acceleration and more complex, variable force scenarios.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key components of Modelling Motion in Unit 4 include:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Advanced Kinematics (Motion Analysis)&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Variable Acceleration:<\/strong> Solving problems where acceleration is not constant, requiring integration of acceleration to find velocity (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">v open paren t close paren<\/annotation><\/semantics><\/math>) and displacement (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">x open paren t close paren<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Alternative Acceleration Formulas:<\/strong> Using expressions for acceleration as a function of velocity (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>f<\/mi><mo>(<\/mo><mi>v<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">a equals f of v<\/annotation><\/semantics><\/math>) or position (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">a equals f of x<\/annotation><\/semantics><\/math>), such as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>v<\/mi><mfrac><mrow><mi>d<\/mi><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"text\/plain\">a equals v d v over d x end-fraction<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Motion Graphs:<\/strong> Interpreting and creating graphs for displacement, velocity, and acceleration.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. Dynamics (Forces and Motion)&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Newton\u2019s Second Law (\ud835\udc39=\ud835\udc5a\ud835\udc4e):<\/strong> Applying the law to both constant and non-constant (variable) forces.<\/li>\n\n\n\n<li><strong>Vector Analysis:<\/strong> Using vector resolution to calculate net force, including horizontal and vertical components, particularly for projectiles and inclined planes.<\/li>\n\n\n\n<li><strong>Momentum and Impulse:<\/strong> Analyzing motion changes using momentum (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mi>m<\/mi><mi>v<\/mi><\/mrow><annotation encoding=\"text\/plain\">p equals m v<\/annotation><\/semantics><\/math>) and impulse.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. Key Topics and Applications&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Simple Harmonic Motion (SHM):<\/strong> Modelling oscillating systems using differential equations, focusing on acceleration <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"text\/plain\">a equals negative n squared x<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Resisted Motion:<\/strong> Modelling motion where forces like air resistance or friction depend on velocity, which requires solving differential equations.<\/li>\n\n\n\n<li><strong>Uniform Circular Motion:<\/strong> Describing motion in terms of a force acting perpendicular to the object&#8217;s velocity.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4. Mathematical Tools&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Differential Equations:<\/strong> Setting up and solving <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">v equals f of x<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>f<\/mi><mo>(<\/mo><mi>v<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">a equals f of v<\/annotation><\/semantics><\/math> equations to model position and speed.<\/li>\n\n\n\n<li><strong>Integration and Differentiation:<\/strong> Used to move between displacement, velocity, and acceleration (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2192<\/mo><mi>v<\/mi><mo>\u2192<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"text\/plain\">x right arrow v right arrow a<\/annotation><\/semantics><\/math>).\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This unit requires students to demonstrate skills in solving complex, often unfamiliar problems related to how objects move in 1D and 2D space, using calculus-based techniques.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u5c02\u9580\u6570\u5b66\uff08\u591a\u304f\u306e\u5834\u5408\u30c8\u30d4\u30c3\u30af4\uff09\u306e\u30e6\u30cb\u30c3\u30c84\u300c\u904b\u52d5\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\u300d\u3067\u306f\u3001\u5fae\u7a4d\u5206\u3068\u30d9\u30af\u30c8\u30eb\u624b\u6cd5\u3092\u7528\u3044\u3066\u7269\u4f53\u306e\u904b\u52d5\u3092\u5206\u6790\u3057\u3001\u904b\u52d5\u5b66\uff08\u904b\u52d5\u306e\u8a18\u8ff0\uff09\u3068\u529b\u5b66\uff08\u904b\u52d5\u3092\u5f15\u304d\u8d77\u3053\u3059\u529b\uff09\u3092\u7d44\u307f\u5408\u308f\u305b\u307e\u3059\u3002\u3053\u306e\u5b66\u7fd2\u3067\u306f\u3001\u3053\u308c\u307e\u3067\u306e\u529b\u5b66\u306e\u77e5\u8b58\u3092\u3001\u4e00\u5b9a\u3067\u306a\u3044\u52a0\u901f\u5ea6\u3084\u3001\u3088\u308a\u8907\u96d1\u3067\u5909\u52d5\u3059\u308b\u529b\u306e\u30b7\u30ca\u30ea\u30aa\u3078\u3068\u62e1\u5f35\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c84\u300c\u904b\u52d5\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\u300d\u306e\u4e3b\u8981\u306a\u69cb\u6210\u8981\u7d20\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u9ad8\u5ea6\u306a\u904b\u52d5\u5b66\uff08\u904b\u52d5\u89e3\u6790\uff09<\/strong><br><strong>\u5909\u52d5\u52a0\u901f\u5ea6<\/strong>\uff1a\u52a0\u901f\u5ea6\u304c\u4e00\u5b9a\u3067\u306f\u306a\u3044\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002\u901f\u5ea6 (<math data-latex=\"v(t)\"><semantics><mrow><mi>v<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>t<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">v(t)<\/annotation><\/semantics><\/math>) \u3068\u5909\u4f4d (<math data-latex=\"x(t)\"><semantics><mrow><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>t<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x(t)<\/annotation><\/semantics><\/math>) \u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u3001\u52a0\u901f\u5ea6\u306e\u7a4d\u5206\u304c\u5fc5\u8981\u3067\u3059\u3002<br><strong>\u52a0\u901f\u5ea6\u306e\u4ee3\u66ff\u516c\u5f0f<\/strong>\uff1a\u52a0\u901f\u5ea6\u3092\u901f\u5ea6 (<math data-latex=\"a=f(v\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a=f(v<\/annotation><\/semantics><\/math>) \u307e\u305f\u306f\u4f4d\u7f6e (<math data-latex=\"a=f(x\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a=f(x<\/annotation><\/semantics><\/math>)) \u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u3059\u5f0f(<math data-latex=\"a=v\\frac{dv}{dx}\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>v<\/mi><mfrac><mrow><mi>d<\/mi><mi>v<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a=v\\frac{dv}{dx}<\/annotation><\/semantics><\/math>) \u306a\u3069  \u3068\u3057\u3066\u8868\u3059\u5f0f\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<br><strong>\u904b\u52d5\u30b0\u30e9\u30d5<\/strong>\uff1a\u5909\u4f4d\u3001\u901f\u5ea6\u3001\u52a0\u901f\u5ea6\u306e\u30b0\u30e9\u30d5\u306e\u89e3\u91c8\u3068\u4f5c\u6210\u3002<\/li>\n\n\n\n<li><strong>\u529b\u5b66\uff08\u529b\u3068\u904b\u52d5\uff09<\/strong><br><strong>\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u7b2c\u4e8c\u6cd5\u5247\uff08<\/strong><math data-latex=\"F=ma\"><semantics><mrow><mi>F<\/mi><mo>=<\/mo><mi>m<\/mi><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">F=ma<\/annotation><\/semantics><\/math>\uff09\uff1a\u3053\u306e\u6cd5\u5247\u3092\u4e00\u5b9a\u529b\u3068\u4e00\u5b9a\u3067\u306a\u3044\uff08\u53ef\u5909\uff09\u529b\u306e\u4e21\u65b9\u306b\u9069\u7528\u3059\u308b\u3002<br><strong>\u30d9\u30af\u30c8\u30eb\u89e3\u6790<\/strong>\uff1a\u30d9\u30af\u30c8\u30eb\u5206\u89e3\u3092\u7528\u3044\u3066\u3001\u7279\u306b\u6295\u5c04\u7269\u3084\u50be\u659c\u9762\u306b\u304a\u3051\u308b\u6c34\u5e73\u6210\u5206\u3068\u5782\u76f4\u6210\u5206\u3092\u542b\u3080\u6b63\u5473\u306e\u529b\u3092\u8a08\u7b97\u3059\u308b\u3002<br><strong>\u904b\u52d5\u91cf\u3068\u529b\u7a4d<\/strong>\uff1a\u904b\u52d5\u91cf\uff08<math data-latex=\"p=mv\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mi>m<\/mi><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p=mv<\/annotation><\/semantics><\/math>\uff09\u3068\u529b\u7a4d\u3092\u7528\u3044\u3066\u904b\u52d5\u306e\u5909\u5316\u3092\u89e3\u6790\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u4e3b\u8981\u30c8\u30d4\u30c3\u30af\u3068\u5fdc\u7528<\/strong><br><strong>\u5358\u632f\u52d5\uff08SHM\uff09<\/strong>\uff1a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u7528\u3044\u3066\u632f\u52d5\u7cfb\u3092\u30e2\u30c7\u30eb\u5316\u3057\u3001\u52a0\u901f\u5ea6 (<math data-latex=\"a=-n^{2}x\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a=-n^{2}x<\/annotation><\/semantics><\/math>) \u306b\u7126\u70b9\u3092\u5f53\u3066\u308b\u3002<br><strong>\u62b5\u6297\u904b\u52d5<\/strong>\uff1a\u7a7a\u6c17\u62b5\u6297\u3084\u6469\u64e6\u306a\u3069\u306e\u529b\u304c\u901f\u5ea6\u306b\u4f9d\u5b58\u3059\u308b\u904b\u52d5\u3092\u30e2\u30c7\u30eb\u5316\u3057\u3001\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u5fc5\u8981\u304c\u3042\u308b\u3002<br><strong>\u7b49\u901f\u5186\u904b\u52d5<\/strong>\uff1a\u7269\u4f53\u306e\u901f\u5ea6\u306b\u5782\u76f4\u306b\u4f5c\u7528\u3059\u308b\u529b\u3092\u7528\u3044\u3066\u904b\u52d5\u3092\u8a18\u8ff0\u3059\u308b\u3002<\/li>\n\n\n\n<li><strong>\u6570\u5b66\u7684\u30c4\u30fc\u30eb<\/strong><br><strong>\u5fae\u5206\u65b9\u7a0b\u5f0f<\/strong>\uff1a\u4f4d\u7f6e\u3068\u901f\u5ea6\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u305f\u3081\u306b\u3001(<math data-latex=\"v=f(x)\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">v=f(x)<\/annotation><\/semantics><\/math>) \u65b9\u7a0b\u5f0f\u3068 (<math data-latex=\"a=f(v)\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>v<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">a=f(v)<\/annotation><\/semantics><\/math>) \u65b9\u7a0b\u5f0f\u3092\u7acb\u3066\u3066\u89e3\u304f\u3002<br><strong>\u7a4d\u5206\u3068\u5fae\u5206<\/strong>\uff1a\u5909\u4f4d\u3001\u901f\u5ea6\u3001\u52a0\u901f\u5ea6 (<math data-latex=\"x\\rightarrow v\\rightarrow a\"><semantics><mrow><mi>x<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi>v<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x\\rightarrow v\\rightarrow a<\/annotation><\/semantics><\/math>) \u9593\u306e\u79fb\u52d5\u306b\u7528\u3044\u308b\u3002<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u5358\u5143\u3067\u306f\u3001\u5fae\u7a4d\u5206\u306b\u57fa\u3065\u304f\u624b\u6cd5\u3092\u7528\u3044\u3066\u30011\u6b21\u5143\u304a\u3088\u30732\u6b21\u5143\u7a7a\u9593\u306b\u304a\u3051\u308b\u7269\u4f53\u306e\u904b\u52d5\u306b\u95a2\u3059\u308b\u8907\u96d1\u3067\u3001\u3057\u3070\u3057\u3070\u99b4\u67d3\u307f\u306e\u306a\u3044\u554f\u984c\u3092\u89e3\u304f\u30b9\u30ad\u30eb\u3092\u751f\u5f92\u304c\u793a\u3059\u3053\u3068\u304c\u6c42\u3081\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Modelling Motion in Unit 4 of QLD Year 12 Specialist Mathematics (often Topic 4) involves using calculus and vector techniques to analyze the movement of objects, combining kinematics (description of motion) and dynamics (forces causing motion). It extends previous mechanics knowledge to non-constant acceleration and more complex, variable force scenarios.&nbsp; Key components of Modelling Motion [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1403","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1403","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1403"}],"version-history":[{"count":5,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1403\/revisions"}],"predecessor-version":[{"id":1412,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1403\/revisions\/1412"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1403"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1403"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1403"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}