document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161528>Question 1161528</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A boy throws a stone from the top of a building 46.0 m above ground. The<BR>\n" );
document.write( "stone is thrown at an angle of 33.0° below the horizontal and strikes the<BR>\n" );
document.write( "ground 55.6 m away from the building, find the following: (a) Time of flight.<BR>\n" );
document.write( "(b) Initial speed. (c) The magnitude and the direction of the velocity of the<BR>\n" );
document.write( "stone just before it strikes the ground. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854443 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5377)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> EDITED for clarity and to correct multiple typos on April 28, 2026:<BR>\n" );
document.write( " <BR>\n" );
document.write( "This is really a Physics class problem,<BR>\n" );
document.write( "but from the math point of view it involves algebra, vectors, geometry and trigonometry.<BR>\n" );
document.write( "Times will be in seconds from the time the stone left the boy's hand,<BR>\n" );
document.write( "distances will be in meters,<BR>\n" );
document.write( "speeds will be in m/s, and accelerations in m/s/s,<BR>\n" );
document.write( "with the units understood, but not always stated.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > initial speed (magnitude of the velocity) of the stone as it leaves the boy's hand<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-10%2C50%2C-30%2C10%2C%0D%0Aarrow%280%2C0%2C39.17%2C-25.43%29%2Clocate%2818%2C-12%2Cv%29%2C%0D%0Agreen%28arrow%280%2C0%2C39.17%2C0%29%29%2Clocate%2825%2C2%2Cgreen%28v%5Bh%5D%29%29%2C%0D%0Agreen%28arrow%2839.17%2C0%2C39.17%2C-25.43%29%29%2Clocate%2840%2C-10%2Cgreen%28v%5Bd%5D%29%29%2C%0D%0Ared%28arc%280%2C0%2C20%2C20%2C0%2C33%29%29%2Clocate%285%2C1%2Cred%2833%5Eo%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bh%5D=v%2Acos33%5Eo ALIGN=MIDDLE  ALT=\"v%5Bh%5D=v%2Acos33%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > magnitude of the horizontal vector component of that initial velocity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bd%5D=v%2Asin33%5Eo ALIGN=MIDDLE  ALT=\"v%5Bd%5D=v%2Asin33%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > magnitude of the vertical, downward vector component of that initial velocity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T ALIGN=MIDDLE  ALT=\"T\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > Time of flight of the stone, in seconds<BR>\n" );
document.write( "We reasonably assume that the effects of air friction or wind do not cause a measurable difference in the motion of the stone,<BR>\n" );
document.write( "so that the horizontal velocity is constant,<BR>\n" );
document.write( "and the vertical velocity is only affected by gravity.<BR>\n" );
document.write( "We know that latitude and altitude affect the acceleration of an object due to gravity,<BR>\n" );
document.write( "but not having precise location data, we will use the value of 9.8m/s/s.<BR>\n" );
document.write( "The downward velocity will increase linearly with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the time in seconds <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the stone was thrown,<BR>\n" );
document.write( "as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2Asin33%5Eo%2B9.8t ALIGN=MIDDLE  ALT=\"v%2Asin33%5Eo%2B9.8t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2Asin33%5Eo ALIGN=MIDDLE  ALT=\"v%2Asin33%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2Asin33%5Eo%2B9.8T ALIGN=MIDDLE  ALT=\"v%2Asin33%5Eo%2B9.8T\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=T ALIGN=MIDDLE  ALT=\"t=T\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The average downward velocity during that time will be the average of initial and final velocities:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28v%2Asin33%5Eo%2Bv%2Asin33%5Eo%2B9.8T%29=v%2Asin33%5Eo%2B4.9T ALIGN=MIDDLE  ALT=\"%281%2F2%29%28v%2Asin33%5Eo%2Bv%2Asin33%5Eo%2B9.8T%29=v%2Asin33%5Eo%2B4.9T\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The horizontal and vertical distances traveled by the stone (in meters) will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28v%2Acos33%5Eo%2AT=55.6%29 ALIGN=MIDDLE  ALT=\"highlight%28v%2Acos33%5Eo%2AT=55.6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28v%2Asin33%5Eo%2B4.9T%29T=46%29 ALIGN=MIDDLE  ALT=\"%28v%2Asin33%5Eo%2B4.9T%29T=46%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28v%2AT%2Asin33%5Eo%2B4.9T%5E2=46%29 ALIGN=MIDDLE  ALT=\"highlight%28v%2AT%2Asin33%5Eo%2B4.9T%5E2=46%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Part (a) and part (b):<BR>\n" );
document.write( "We can find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T ALIGN=MIDDLE  ALT=\"T\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the equations above.<BR>\n" );
document.write( "From the first of the highlighted equations above, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2AT=55.6%281%2Fcos33%5Eo%29 ALIGN=MIDDLE  ALT=\"v%2AT=55.6%281%2Fcos33%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and substituting the expression found for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2AT ALIGN=MIDDLE  ALT=\"v%2AT\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the second equation, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=55.6%281%2Fcos33%5Eo%29%2Asin33%5Eo%2B4.9T%5E2=46 ALIGN=MIDDLE  ALT=\"55.6%281%2Fcos33%5Eo%29%2Asin33%5Eo%2B4.9T%5E2=46\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=55.6%28sin33%5Eo%2Fcos33%5Eo%29%2B4.9T%5E2=46 ALIGN=MIDDLE  ALT=\"55.6%28sin33%5Eo%2Fcos33%5Eo%29%2B4.9T%5E2=46\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=55.6%2Atan33%5Eo%2B4.9T%5E2=46 ALIGN=MIDDLE  ALT=\"55.6%2Atan33%5Eo%2B4.9T%5E2=46\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan33%5Eo=0.64941 ALIGN=MIDDLE  ALT=\"tan33%5Eo=0.64941\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded) , we substitute to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=55.6%2A0.64941%2B4.9T%5E2=46 ALIGN=MIDDLE  ALT=\"55.6%2A0.64941%2B4.9T%5E2=46\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5E2=%2846-55.6%2A0.64941%29%2F4.9=2.0189 ALIGN=MIDDLE  ALT=\"T%5E2=%2846-55.6%2A0.64941%29%2F4.9=2.0189\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded), and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T=sqrt%282.0189%29 ALIGN=MIDDLE  ALT=\"T=sqrt%282.0189%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28T=1.42%29 ALIGN=MIDDLE  ALT=\"highlight%28T=1.42%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "Answer for part (a): The stone flies for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28T=1.42seconds%29 ALIGN=MIDDLE  ALT=\"highlight%28T=1.42seconds%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, substituting the value found above and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos33%5Eo=0.83867 ALIGN=MIDDLE  ALT=\"cos33%5Eo=0.83867\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28v%2Acos33%5Eo%2AT=55.6%29 ALIGN=MIDDLE  ALT=\"highlight%28v%2Acos33%5Eo%2AT=55.6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2A0.83867%2A1.42=55.6 ALIGN=MIDDLE  ALT=\"v%2A0.83867%2A1.42=55.6\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=55.6%2F1.42%2F0.83867 ALIGN=MIDDLE  ALT=\"v=55.6%2F1.42%2F0.83867\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28v=46.7%29 ALIGN=MIDDLE  ALT=\"highlight%28v=46.7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded). <BR>\n" );
document.write( "The initial speed (magnitude of the initial velocity) of the stone was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=46.7 ALIGN=MIDDLE  ALT=\"46.7\"   DISABLED_style= \"max-width:90%; height:auto;\" > m/s.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Part(c):<BR>\n" );
document.write( "For the stone, the time \"just before it strikes the ground\" is a minuscule fraction of a second before t=1.42 seconds, which rounds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T=1.42seconds ALIGN=MIDDLE  ALT=\"T=1.42seconds\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can calculate the magnitude and the direction of the velocity of the<BR>\n" );
document.write( "stone at that time from the magnitude of its horizontal and downward components.<BR>\n" );
document.write( "As stated early,<BR>\n" );
document.write( "the horizontal component of the stone velocity is constant, with magnitude <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bh%5D=v%2Acos33%5Eo ALIGN=MIDDLE  ALT=\"v%5Bh%5D=v%2Acos33%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "while the vertical component is a function of the time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , in seconds,  since the stone was thrown. <BR>\n" );
document.write( "It was also established that at time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T=1.42seconds ALIGN=MIDDLE  ALT=\"T=1.42seconds\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the downwards velocity was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bd%5D=v%2Asin33%5Eo%2B9.8T ALIGN=MIDDLE  ALT=\"V%5Bd%5D=v%2Asin33%5Eo%2B9.8T\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos33%5Eo=0.83867 ALIGN=MIDDLE  ALT=\"cos33%5Eo=0.83867\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin33%5Eo=0.54464 ALIGN=MIDDLE  ALT=\"sin33%5Eo=0.54464\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the already found values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T=1.42 ALIGN=MIDDLE  ALT=\"T=1.42\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=46.7 ALIGN=MIDDLE  ALT=\"v=46.7\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bh%5D=46.7%2A0.83867=39.17 ALIGN=MIDDLE  ALT=\"v%5Bh%5D=46.7%2A0.83867=39.17\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5Bd%5D=46.7%2A0.54464%5Eo%2B9.8%2A1.42=39.35 ALIGN=MIDDLE  ALT=\"V%5Bd%5D=46.7%2A0.54464%5Eo%2B9.8%2A1.42=39.35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C250%2C-10%2C50%2C-45%2C5%2C%0D%0Aarrow%280%2C0%2C39.17%2C-39.35%29%2Clocate%2818%2C-19%2CV%29%2C%0D%0Agreen%28arrow%280%2C0%2C39.17%2C0%29%29%2Clocate%2825%2C2%2Cgreen%28v%5Bh%5D%29%29%2C%0D%0Agreen%28arrow%2839.17%2C0%2C39.17%2C-39.35%29%29%2Clocate%2840%2C-10%2Cgreen%28V%5Bd%5D%29%29%2C%0D%0Ared%28arc%280%2C0%2C20%2C20%2C0%2C45.1%29%29%2Clocate%285%2C-0.5%2Cred%28theta%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding those horizontal and vertical components vectors,<BR>\n" );
document.write( "we calculate the magnitude of the velocity vector at 1.42 second as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=sqrt%2839.17%5E2%2B39.35%5E2%29 ALIGN=MIDDLE  ALT=\"V=sqrt%2839.17%5E2%2B39.35%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28V=67.9%29 ALIGN=MIDDLE  ALT=\"highlight%28V=67.9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can calculate its direction as the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > below the horizontal from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=39.35%2F39.17=1.0046 ALIGN=MIDDLE  ALT=\"tan%28theta%29=39.35%2F39.17=1.0046\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded), which corresponds to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=45.1%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=45.1%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The calculated magnitude and direction of the velocity of the<BR>\n" );
document.write( "stone just before it strikes the ground is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2867.9%29 ALIGN=MIDDLE  ALT=\"highlight%2867.9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >m/s at an angle below the horizontal of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2845.1%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%2845.1%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );