Classical Mechanics - Orbital Path

When it is about the same distance from the Sun as is Jupiter, a spacecraft on a mission to the outer planets has a speed that is 1.5 times the speed of Jupiter in its orbit. Which of the following describes the orbit of the spacecraft about the Sun?

A. Spiral
B. Circle
C. Ellipse
D. Parabola
E. Hyperbola

(GR9677 #66)

Solution:

The mission is to outer planets, so the path should not be bounded (no longer a close path): A, B, C are FALSE.

If v_escape = √2 v_circular → Parabola
If v_e  √2 v_c → Hyperbola
Since v_e = 1.5  √2 → Hyperbola

Answer: E

Notes:

To find escape velocity:
F_c = F_g 
mv²/ r = GMm / r² 
v_circular = √(GM / r)

KE = PE_gravity 
½ mv² = GMm / r  
v_escape = √(2GM / r) = √2 √(GM / r)
v_escape = √2 v_circular