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as possible. Never leave the answer in terms of sin or cos /2; convert the halfangle to phi

(C) Determine x(phi) and y( phi).style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.214em; vertical-align: -0.277em;" data-mce-style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.214em; vertical-align: -0.277em;">style="display: inline-block; width: 0px; height: 1.988em;" data-mce-style="display: inline-block; width: 0px; height: 1.988em;">style="font-family: helvetica, sans-serif; font-size: 12px;" data-mce-style="font-family: helvetica, sans-serif; font-size: 12px;">class="mi" id="mathjax-span-6" data-mce-style="font-family: mathjax_math; font-style: italic;">class="mrow" id="mathjax-span-5" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);" data-mce-style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);">style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;" data-mce-style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;">class="math" id="mathjax-span-4" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;" data-mce-style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;">class="mathjax" role="textbox" aria-readonly="true" data-mce-style="font-size: 12px; word-spacing: 0px;">style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.214em; vertical-align: -0.277em;" data-mce-style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.214em; vertical-align: -0.277em;">style="display: inline-block; width: 0px; height: 1.988em;" data-mce-style="display: inline-block; width: 0px; height: 1.988em;">class="mi" id="mathjax-span-6" data-mce-style="font-family: mathjax_math; font-style: italic;">class="mrow" id="mathjax-span-5" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);" data-mce-style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);">style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;" data-mce-style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;">class="math" id="mathjax-span-4" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;" data-mce-style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;">class="mathjax" role="textbox" aria-readonly="true" data-mce-style="font-size: 12px; word-spacing: 0px;">

(D) Determine xi( phi) and eta ( phistyle="font-family: helvetica, sans-serif; font-size: 12px;" data-mce-style="font-family: helvetica, sans-serif; font-size: 12px;">class="mi" id="mathjax-span-6" data-mce-style="font-family: mathjax_math; font-style: italic;">class="mrow" id="mathjax-span-5" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);" data-mce-style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);">style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;" data-mce-style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;">class="math" id="mathjax-span-4" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;" data-mce-style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;">class="mathjax" role="textbox" aria-readonly="true" data-mce-style="font-size: 12px; word-spacing: 0px;">style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.214em; vertical-align: -0.277em;" data-mce-style="border-left-width: 0em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.214em; vertical-align: -0.277em;">style="display: inline-block; width: 0px; height: 1.988em;" data-mce-style="display: inline-block; width: 0px; height: 1.988em;">class="mi" id="mathjax-span-6" data-mce-style="font-family: mathjax_math; font-style: italic;">class="mrow" id="mathjax-span-5" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);" data-mce-style="position: absolute; top: -1.988em; left: 0em; clip: rect(1.108em 1000em 2.378em -0.535em);">style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;" data-mce-style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 135%;">class="math" id="mathjax-span-4" data-mce-style="display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px;">style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;" data-mce-style="border: 0px; padding: 0px; margin: 0px; max-width: none; max-height: none; vertical-align: 0px;">class="mathjax" role="textbox" aria-readonly="true" data-mce-style="font-size: 12px; word-spacing: 0px;"> ).

(E) Express r as a function of .

(F) The angular momentum is L = m r

2

d theta /dt. Determine L in terms of {a,e,T}

Hence verify that L is a constant of the motion.

(G) The energy is

E = [1/2] m ( dr/dt )^2 + [1/2] m r^2 (dtheta/dt)^2 - GMm /r.

Determine E in terms of {a,e,T}.

(H) E must be a constant of the motion. Hence determine T and E from the

equation that you obtained in (G).

(I) Write T in terms of the spatial orbit parameters and the force constant GM.

(J) Now express L and E in terms of a, e, GM.

(K) In Figure 1, determine the Cartesian coordinates of the points P, A, and R

(L) In Figure 1, determine the time t at the point R.

Determine x(phi ) Determine xi( phi) and eta ( phi ) Express r as a function of . The angular momentum is L = m r d theta /dt. Determine L in terms of {a,e,T} Hence verify that L is a constant of the motion. The energy is E = [1/2] m ( dr/dt )^2 + [1/2] m r^2 (d theta/dt)^2 - GMm /r. Determine E in terms of {a,e,T}. E must be a constant of the motion. Hence determine T and E from the equation that you obtained in (G). Write T in terms of the spatial orbit parameters and the force constant GM. Now express L and E in terms of a, e, GM. In Figure 1, determine the Cartesian coordinates of the points P, A, and R In Figure 1, determine the time t at the point R.

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