physics problemo???

Question

(a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth's orbit?
(b) Voyager 1 achieved a maximum speed of 125,000 km/h on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?

Answer 1

To compute escape velocity we are equating the potential energy Pe an object of mass m can have if achieved an orbit of radius R around a mass M to the kinetic energy Ke it could achieve if allowed to free fall back to body M.

Ke=Pe
0.5 m V^2 = FR
F= GMm/R^2 we have
0.5 m V^2 = (GMm/R^2 )R
0.5 m V^2 = GMm/R
V= sqrt(2GMm/(Rm))
V= sqrt(2GM/R)

G- gravitational constant = 6.673 E-11 m3/ (kg s^2)
M- mass of the Sun= 1.9891 E+30 kg
R- radius of the solar system = 7.5x10^15m (He-he...enough to hit Kuiper Belt 30,000 AU and the Oort Cloud 50,000AU. The Oort Cloud has a larger radius, estimated at about 50,000 AU (or 7.5x10^15 m).
Now we are ready

V= sqrt(2GM/R)
V= sqrt(2x6.673 E-11 x 1.9891 E+30 /7.5x10^15 )
V=188 m/s
V= 680 km/h
If you are looking to establish an orbit around the Sun escape velocity of of 620,000m/s would do just fine. The escape velocity is inversely proportional to the ratio of orbit you want to achieve.

"To leave planet Earth an escape velocity of 11.2 km/s is required, however a speed of 42.1 km/s is required to escape the Sun's gravity (and exit the solar system) from the same position"

Answer 2

42.1 for those WA people