To determine the speed of a planet at its furthest approach in an elliptical orbit, the conservation of mechanical energy principle can be applied. The total mechanical energy remains constant, meaning the sum of kinetic and potential energy at the closest approach equals that at the furthest point. The relationship between speeds at different distances can be simplified using the ratio of distances, leading to the conclusion that speed decreases as distance increases. The key to solving the problem lies in recognizing that the gravitational potential energy changes inversely with distance, affecting the kinetic energy accordingly. Understanding these principles allows for a more straightforward calculation of the planet's speed at varying distances in its orbit.