Question

Problem #1 Consider a follow-on mission to DART. In the new mission, engineers have to decided to now crash a spacecraft into the larger of the two asteroids, Didymos. For your analysis, you will ignore the effects of Didymos's moon, Dimorphos. Assume that Didymos is approximately spherical in shape with radius, $R_{Didymos} = 389.99$ [m]. The grav- itational parameter of Didymos is $\mu = 34.71$ [$m^3/s^2$]. The spacecraft is on a hyperbolic approach with respect to Didymos and has a hyperbolic exceess velocity $v_\infty = 0.25$ [m/s]. a) At some distance $r_0 = 340 \cdot R_{Didymos}$, your navigation computer reports that your flight path angle is $\gamma = -89.5^\circ$. What is your true anomaly, $\nu_0$ at this point? Show all steps by hand using symbolic expressions, providing numerical values for intermediate terms along the way. b) Will your spacecraft successfully collide with Didymos? Show all steps by hand using symbolic expressions, providing numerical values for intermediate terms along the way. c) Plot the trajectory in Cartesian coordinates to confirm your result. Provide two plots: i. The full trajectory starting at $\nu_0$ and ending and $\nu_f = \nu_0 + \pi$. ii. A \"zoomed in\" version of the previous plot. On this same plot, show the extent of the asteroid by plotting a circle of radius $R_{Didymos}$. Please review the homework submission guidelines on Brightspace! As a reminder \"Your written work must still include a full description of the methods and equations used to solve the problem or make the plot; the phrase 'refer to my code' is never an adequate substitute.\"

          Problem #1
Consider a follow-on mission to DART. In the new mission, engineers have to decided to now crash a
spacecraft into the larger of the two asteroids, Didymos. For your analysis, you will ignore the effects of
Didymos's moon, Dimorphos.
Assume that Didymos is approximately spherical in shape with radius, $R_{Didymos} = 389.99$ [m]. The grav-
itational parameter of Didymos is $\mu = 34.71$ [$m^3/s^2$]. The spacecraft is on a hyperbolic approach with
respect to Didymos and has a hyperbolic exceess velocity $v_\infty = 0.25$ [m/s].
a) At some distance $r_0 = 340 \cdot R_{Didymos}$, your navigation computer reports that your flight path angle
is $\gamma = -89.5^\circ$. What is your true anomaly, $\nu_0$ at this point? Show all steps by hand using symbolic
expressions, providing numerical values for intermediate terms along the way.
b) Will your spacecraft successfully collide with Didymos? Show all steps by hand using symbolic
expressions, providing numerical values for intermediate terms along the way.
c) Plot the trajectory in Cartesian coordinates to confirm your result. Provide two plots:
i. The full trajectory starting at $\nu_0$ and ending and $\nu_f = \nu_0 + \pi$.
ii. A \"zoomed in\" version of the previous plot. On this same plot, show the extent of the asteroid
by plotting a circle of radius $R_{Didymos}$.
Please review the homework submission guidelines on Brightspace! As a reminder \"Your written work
must still include a full description of the methods and equations used to solve the problem or make
the plot; the phrase 'refer to my code' is never an adequate substitute.\"

$Problem #1 Consider a follow-on mission to DART. In the new mission, engineers have to decided to now crash a spacecraft into the larger of the two asteroids, Didymos. For your analysis, you will ignore the effects of Didymos's moon, Dimorphos. Assume that Didymos is approximately spherical in shape with radius, $R{Didymos} = 389.99$ [m]. The grav- itational parameter of Didymos is $\mu = 34.71$ [$m^3/s^2$]. The spacecraft is on a hyperbolic approach with respect to Didymos and has a hyperbolic exceess velocity $v\infty = 0.25$ [m/s]. a) At some distance $r0 = 340 \cdot R{Didymos}$, your navigation computer reports that your flight path angle is $\gamma = -89.5^\circ$. What is your true anomaly, $\nu0$ at this point? Show all steps by hand using symbolic expressions, providing numerical values for intermediate terms along the way. b) Will your spacecraft successfully collide with Didymos? Show all steps by hand using symbolic expressions, providing numerical values for intermediate terms along the way. c) Plot the trajectory in Cartesian coordinates to confirm your result. Provide two plots: i. The full trajectory starting at $\nu0$ and ending and $\nuf = \nu0 + \pi$. ii. A \"zoomed in\" version of the previous plot. On this same plot, show the extent of the asteroid by plotting a circle of radius $R{Didymos}$. Please review the homework submission guidelines on Brightspace! As a reminder \"Your written work must still include a full description of the methods and equations used to solve the problem or make the plot; the phrase 'refer to my code' is never an adequate substitute.\"$

Added by Gary G.

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