Final answer:
To find the speed at aphelion, we use the conservation of angular momentum in planetary motion, which gives us a speed of 21.43 km/s when substituting the given values.
Step-by-step explanation:
To determine the speed of a planet at aphelion, we can apply the conservation of angular momentum, which states that the angular momentum of a system remains constant if there is no net external torque. In an elliptical orbit, a planet's angular momentum is conserved, and thus the product of its speed and the radial distance from the focus (the star) is constant at any two points of the orbit. This means we can set up the following equation:
m × v_perihelion × r_perihelion = m × v_aphelion × r_aphelion
Where:
- m is the mass of the planet (which cancels out)
- v_perihelion is the speed at perihelion
- r_perihelion is the radial distance at perihelion
- v_aphelion is the speed at aphelion
- r_aphelion is the radial distance at aphelion
Given that v_perihelion = 30 km/s and r_perihelion = 2.50 × 10^8 km, r_aphelion = 3.50 × 10^8 km, we can solve for v_aphelion:
v_aphelion = (v_perihelion × r_perihelion) / r_aphelion
After substituting the values:
v_aphelion = (30 km/s × 2.50 × 10^8 km) / (3.50 × 10^8 km)
v_aphelion = 21.43 km/s
Therefore, the speed of the planet at aphelion is 21.43 km/s.