A spacecraft of mass m is first brought into an orbit around the earth. The earth (together with the spacecraft) orbits the sun in a near circular orbit with radius R (R is the mean distance between the earth and the sun; it is about 150 million km).

(a) What is the speed v0 (in m/s) of the earth in its orbit of radius R=1.5×1011 m around the sun with a mass M=1.99×1030 kg? Take the gravitational constant G=6.674×10−11 m3kg−1s−2.

problem#8.3 from assignment#8-MIT

To find the speed of the earth in its orbit, we can use the concept of gravitational force between the sun and the earth.

The force of gravity between two objects can be calculated using the formula:

F = (G * M * m) / r^2

Where:
F is the force of gravity
G is the gravitational constant
M is the mass of one object (in this case, the mass of the sun)
m is the mass of the other object (in this case, the mass of the earth)
r is the distance between the centers of the two objects (in this case, the radius of the earth's orbit)

The force of gravity provides the centripetal force that keeps the earth in its orbit. The centripetal force can be calculated using the formula:

F = (m * v^2) / r

Where:
F is the centripetal force
m is the mass of the object (in this case, the mass of the earth)
v is the speed of the object in its orbit
r is the radius of the orbit

Setting the equations for gravitational force and centripetal force equal to each other, we get:

(G * M * m) / r^2 = (m * v^2) / r

Simplifying and solving for v, we get:

v = sqrt((G * M) / r)

Now we can substitute the given values into the equation:

G = 6.674 × 10^-11 m^3kg^-1s^-2
M = 1.99 × 10^30 kg
r = 1.5 × 10^11 m

Plugging these values into the equation:

v = sqrt((6.674 × 10^-11 m^3kg^-1s^-2 * 1.99 × 10^30 kg) / (1.5 × 10^11 m))

Evaluating the expression:

v ≈ 29,457 m/s

Therefore, the speed of the earth in its orbit around the sun is approximately 29,457 m/s.