Question

The first artificial satellite, Sputnik 1, was a 58 cm diameter, 83.6 kg sphere launched into an elliptical orbit by the Soviet Union in 1957. At the perigee (closest point) of its orbit, its distance from Earth’s center was 6.81e6 m, while at the apogee (farthest point), its distance was 7.53e6 m. (Re = 6.37e6 m, Me = 5.97e24 kg, G = 6.67e-11 N-m²/kg²)

a) What was its average speed, approximating the elliptical orbit as a circle with the radius an average of the two distances?

b) How much energy was required to put it into that orbit from its launch pad in the Kazakh Soviet Socialist Republic?

Answer 1

The average speed of Sputnik 1 in its elliptical orbit is approximately 7631 m/s. The energy required to put the satellite into its orbit is approximately 8.86e8 J.

Step-by-step explanation:

To find the average speed of the artificial satellite Sputnik 1, we can approximate its elliptical orbit as a circle with the radius being the average of the perigee and apogee distances. The perigee distance is 6.81e6 m and the apogee distance is 7.53e6 m. The average radius would be (6.81e6 + 7.53e6) / 2 = 7.17e6 m.

To calculate the average speed, we need to use the formula v = sqrt(G * Me / R), where G is the gravitational constant (6.67e-11 N-m²/kg²), Me is the mass of the Earth (5.97e24 kg), and R is the average radius of the orbit.

Plugging in the values, we get v = sqrt((6.67e-11 N-m²/kg²) * (5.97e24 kg) / (7.17e6 m)) ≈ 7631 m/s. Therefore, the average speed of Sputnik 1 in its elliptical orbit is approximately 7631 m/s.

To calculate the energy required to put Sputnik 1 into its orbit, we can use the formula E = - (G * Me * Ms) / (2 * R), where Ms is the mass of Sputnik 1 (83.6 kg) and R is the average radius of the orbit.

Plugging in the values, we get E = - ((6.67e-11 N-m²/kg²) * (5.97e24 kg) * (83.6 kg)) / (2 * 7.17e6 m) ≈ - 8.86e8 J. Therefore, the energy required to put Sputnik 1 into its orbit from its launch pad is approximately 8.86e8 J.