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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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Projectile Motion Find the tangential and normal components of acceleration for a projectile fired at an angle θ with the horizontal at an initial speed of v 0 . What are the components when the projectile is at its maximum height?

To determine

To Calculate: The normal and tangential components of acceleration for a projectile fired an angle θ with respect to the horizontal at an initial speed of vo. Also calculate the normal and tangential components when the projectile is at its maximum height.

Explanation

Given: Projectile is fired at an angle θ with respect to the horizontal component at an initial speed of vo

Formula Used:

if r(t)=a i+b j+c k||r(t)||=a2+b2+c2an=v×avaT=a.T

Calculation: From theorem position vector of a projectile, we understand that: r(t)=(v0tcosθ)i + (h +v0tsinθ16t2)jThen the velocity vector , v(t)=(v0cosθ)i+(v0sinθ32t)jThe acceleration vector, a(t) = -32jThe unit tangent vector T(t)= v(t)v(t)=v0cosθi+(v0sinθ32t)jv02cos2θ+(v0sinθ32t)2=v0cosθi+(v0sinθ32t)jv0264t v0sinθ+(32t)2differentiate,T(t),  with respect to t,      we getT|(t)=32jv0264t v0sinθ+(32t)2(64v0sinθ+32×32.2.t)[v0cosθi+(v0sinθ32t)]j2(v0264tv0sinθ+(32t)2)32=32(v0264t v0sinθ+(32t)2)j32(v0sinθ+32t)[v0cosθi+(v0sinθ32t)]j(v0264tv0s

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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 P-10PSCh-12 P-11PSCh-12 P-12PSCh-12 P-13PSCh-12 P-14PS

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