document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=356762>Question 356762</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An unidentified object moves along the s-axis, with displacement s = s(t) (meters), velocity v = v(t) (m/sec), and acceleration a = a(t) (m/sē). It so happens that the velocity and displacement are related by the equation v = sqrt(8s+16). Moreover, at the instant t = 0, the object is observed at s = 6. A) Show that a is constant, and show its value. B) Graph v as a function of s. C) Graph v as a function of t. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #254669 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=254669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  v=Sqrt(8s+16), <BR>\n" );
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document.write( "A) a=dv/dt=1/2*1/Sqrt(8s+16)*8v=4v/Sqrt(8s+16)=4v/v=4 (m/s^2)<BR>\n" );
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document.write( " (Other way : v^2*m/2=Kinetic Energy=m*(4s+8)<BR>\n" );
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document.write( " Potential Energy = Constant - Kinetic Energy = Const - 4ms -8m<BR>\n" );
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document.write( " Force = - d(Potential Energy)/ds = 4m = m*a => a=4 m/s^2)<BR>\n" );
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document.write( "B) v=Sqrt(8s+16)<BR>\n" );
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document.write( "with v=ds/dt we get a differential equation : <BR>\n" );
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document.write( "ds/Sqrt(8s+16)=dt=ds/(2Sqrt(2s+4))<BR>\n" );
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document.write( "Integration gives : <BR>\n" );
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document.write( " t+C=1/2*Sqrt(2s+4), <BR>\n" );
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document.write( " indeed d(t+C)=dt, C is a constant<BR>\n" );
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document.write( " 1/2*d(Sqrt(2s+4))=1/2*1/(2Sqrt(2s+4))*2=1/(2*Sqrt(2s+4))=1/Sqrt(8s+16)<BR>\n" );
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document.write( " s(t=0)=6 : at t=0, s=6, in the equation : C=1/2*Sqrt(12+4)=1/2*Sqrt(16)=2<BR>\n" );
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document.write( "the function displacement towards time is : <BR>\n" );
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document.write( " s(t)=1/2*(2t+4)^2-4<BR>\n" );
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document.write( "   =1/2(4t^2+16*t+16)-16<BR>\n" );
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document.write( "   =2t^2+8t-8 (m)<BR>\n" );
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document.write( "C) the velocity is :  v(t)=ds/dt= <BR>\n" );
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document.write( "   =4t+8 (m/s)<BR>\n" );
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document.write( "Verification : acceleration is : a(t)=dv/dt=4 (m/s^2)<BR>\n" );
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