{"id":195460,"date":"2025-02-28T09:29:54","date_gmt":"2025-02-28T09:29:54","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195460"},"modified":"2025-02-28T09:29:57","modified_gmt":"2025-02-28T09:29:57","slug":"a-ball-is-to-be-shot-from-level-ground-with-a-certain-speed","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/28\/a-ball-is-to-be-shot-from-level-ground-with-a-certain-speed\/","title":{"rendered":"A ball is to be shot from level ground with a certain speed"},"content":{"rendered":"\n

A ball is to be shot from level ground with a certain speed. Figure 4-45 shows the range R it will have versus the launch angle u0. The value of u0 determines the flight time; let tmax represent the maximum flight time. What is the least speed the ball will have during its flight if u0 is chosen such that the flight time is 0.500tmax?<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

In projectile motion, the flight time ( T ) of a projectile launched from and landing on level ground is given by:<\/p>\n\n\n\n

[ T = \\frac{2 v_0 \\sin \\theta_0}{g} ]<\/p>\n\n\n\n

where ( v_0 ) is the initial launch speed, ( \\theta_0 ) is the launch angle, and ( g ) is the acceleration due to gravity. The maximum flight time, ( T_{\\text{max}} ), occurs at a launch angle of 90\u00b0, resulting in:<\/p>\n\n\n\n

[ T_{\\text{max}} = \\frac{2 v_0}{g} ]<\/p>\n\n\n\n

Given that the desired flight time is half of the maximum, we have:<\/p>\n\n\n\n

[ T = 0.5 \\times T_{\\text{max}} = \\frac{v_0}{g} ]<\/p>\n\n\n\n

Equating the two expressions for ( T ):<\/p>\n\n\n\n

[ \\frac{2 v_0 \\sin \\theta_0}{g} = \\frac{v_0}{g} ]<\/p>\n\n\n\n

Simplifying, we find:<\/p>\n\n\n\n

[ 2 \\sin \\theta_0 = 1 ]<\/p>\n\n\n\n

[ \\sin \\theta_0 = \\frac{1}{2} ]<\/p>\n\n\n\n

Therefore:<\/p>\n\n\n\n

[ \\theta_0 = \\sin^{-1}\\left(\\frac{1}{2}\\right) = 30^\\circ ]<\/p>\n\n\n\n

The horizontal and vertical components of the velocity at launch are:<\/p>\n\n\n\n

[ v_{0x} = v_0 \\cos \\theta_0 = v_0 \\cos 30^\\circ = v_0 \\times \\frac{\\sqrt{3}}{2} ]<\/p>\n\n\n\n

[ v_{0y} = v_0 \\sin \\theta_0 = v_0 \\sin 30^\\circ = v_0 \\times \\frac{1}{2} ]<\/p>\n\n\n\n

At the peak of its trajectory, the vertical component of the velocity becomes zero, leaving only the horizontal component. Thus, the minimum speed of the ball during its flight is:<\/p>\n\n\n\n

[ v_{\\text{min}} = v_{0x} = v_0 \\times \\frac{\\sqrt{3}}{2} ]<\/p>\n\n\n\n

This result indicates that when the flight time is half of the maximum possible, the ball reaches its minimum speed at the peak of its trajectory, which is ( \\frac{\\sqrt{3}}{2} ) (approximately 0.866) times the initial launch speed.<\/p>\n\n\n\n

This analysis demonstrates the relationship between launch angle, flight time, and speed in projectile motion. By selecting a launch angle of 30\u00b0, the projectile achieves a flight time that is half of the maximum, and its minimum speed during flight occurs at the apex, determined solely by the horizontal component of the initial velocity.<\/p>\n","protected":false},"excerpt":{"rendered":"

A ball is to be shot from level ground with a certain speed. Figure 4-45 shows the range R it will have versus the launch angle u0. The value of u0 determines the flight time; let tmax represent the maximum flight time. What is the least speed the ball will have during its flight if […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195460","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195460"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195460\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195460"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}