A building is 50 meters tall. If you drop a penny from the roof, its height after t seconds is given s(t)=100-4.9t2 in meters (until it hits the ground)<\/p>\n\n\n\n
a) Compute the velocity of the penny after t seconds (before it hits the ground)<\/p>\n\n\n\n
b) Compute the acceleration of the penny after t seconds (before it hits the ground).<\/p>\n\n\n\n
c) At which time does the penny hit the ground?<\/p>\n\n\n\n
d) What is the velocity of the penny when it hits the ground?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n Let’s go through each part of the problem step by step.<\/p>\n\n\n\n The height of the penny after ( t ) seconds is given by the equation: Velocity is the rate of change of the object’s position with respect to time. To find the velocity function ( v(t) ), we need to take the derivative of the height function ( s(t) ) with respect to time:<\/p>\n\n\n\n [ [ Acceleration is the rate of change of velocity with respect to time. To find the acceleration ( a(t) ), we take the derivative of the velocity function ( v(t) ) with respect to time:<\/p>\n\n\n\n [ [ The penny hits the ground when its height ( s(t) ) reaches 0 meters. So, we set the height function equal to 0 and solve for ( t ):<\/p>\n\n\n\n [ To find the velocity when the penny hits the ground, we substitute ( t = 4.51 ) seconds into the velocity function ( v(t) = -9.8t ):<\/p>\n\n\n\n [ This problem demonstrates the basic principles of motion under gravity, with the penny falling from a height and undergoing constant acceleration due to Earth’s gravitational pull.<\/p>\n","protected":false},"excerpt":{"rendered":" A building is 50 meters tall. If you drop a penny from the roof, its height after t seconds is given s(t)=100-4.9t2 in meters (until it hits the ground) a) Compute the velocity of the penny after t seconds (before it hits the ground) b) Compute the acceleration of the penny after t seconds (before […]<\/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-188655","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188655","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=188655"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188655\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188655"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188655"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Given Information:<\/h3>\n\n\n\n
[
s(t) = 100 – 4.9t^2
]
This is a quadratic equation that describes the motion of an object under the influence of gravity. The equation tells us that at ( t = 0 ), the penny starts from a height of 100 meters, and its height decreases as time progresses due to the acceleration of gravity.<\/p>\n\n\n\nPart a) Compute the velocity of the penny after ( t ) seconds.<\/h3>\n\n\n\n
v(t) = \\frac{ds(t)}{dt}
]
Let’s compute this derivative:<\/p>\n\n\n\n
v(t) = \\frac{d}{dt}(100 – 4.9t^2)
]
The derivative of 100 is 0, and the derivative of ( -4.9t^2 ) is ( -9.8t ). Thus:
[
v(t) = -9.8t
]
This equation tells us that the velocity of the penny is proportional to the time, with the negative sign indicating that the penny is falling downward.<\/p>\n\n\n\nPart b) Compute the acceleration of the penny after ( t ) seconds.<\/h3>\n\n\n\n
a(t) = \\frac{dv(t)}{dt}
]
Since ( v(t) = -9.8t ), its derivative is:<\/p>\n\n\n\n
a(t) = -9.8
]
This shows that the acceleration of the penny is constant at ( -9.8 \\, \\text{m\/s}^2 ), which is the acceleration due to gravity near the Earth’s surface. The negative sign indicates that the acceleration is downward.<\/p>\n\n\n\nPart c) At which time does the penny hit the ground?<\/h3>\n\n\n\n
s(t) = 100 – 4.9t^2 = 0
]
Solving for ( t ):
[
4.9t^2 = 100
]
[
t^2 = \\frac{100}{4.9}
]
[
t^2 \\approx 20.41
]
[
t \\approx \\sqrt{20.41} \\approx 4.51 \\, \\text{seconds}
]
Thus, the penny hits the ground at approximately ( t \\approx 4.51 ) seconds.<\/p>\n\n\n\nPart d) What is the velocity of the penny when it hits the ground?<\/h3>\n\n\n\n
v(4.51) = -9.8 \\times 4.51 \\approx -44.95 \\, \\text{m\/s}
]
Thus, the velocity of the penny when it hits the ground is approximately ( -44.95 \\, \\text{m\/s} ), indicating that the penny is falling downward at a speed of about 44.95 meters per second.<\/p>\n\n\n\nSummary of Answers:<\/h3>\n\n\n\n
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