Question

Scientists have proposed using the radiation pressure of sunlight for travel to other planets in the Solar System. If the intensity of the electromagnetic radiation produced by the Sun is about \(1.40 \mathrm{~kW} / \mathrm{m}^{2}\) near the Earth, what size would a sail have to be to accelerate a spaceship with a mass of 10.0 metric tons at \(1.00 \mathrm{~m} / \mathrm{s}^{2} ?\) a) Assume that the sail absorbs all the incident radiation. b) Assume that the sail perfectly reflects all the incident radiation.

Short answer

Answer: When the sail absorbs all the incident radiation, the required sail area is approximately 2.14 x 10^12 m^2. When the sail perfectly reflects all the incident radiation, the required sail area is approximately 1.07 x 10^12 m^2.

Step-by-step solution

a) Sail absorbs all the incident radiation

: Step 1: Calculate force required to accelerate the spaceship F = ma F = (10.0 * 1000 kg)(1.00 m/s^2) // convert 10.0 metric tons to kg F = 10,000 N Step 2: Calculate radiation pressure when absorbed Radiation pressure (P) when absorbed can be found using the formula: P = (Intensity of sunlight)/(Speed of light*c), where c = 3.00 x 10^8 m/s P = (1.40 kW/m^2 * 1,000 W/kW) / (3.00 * 10^8 m/s) P = 4.67 * 10^(-9) N/m^2 Step 3: Calculate sail area required Area = Force / Radiation pressure Area = 10,000 N / 4.67 * 10^(-9) N/m^2 Area ≈ 2.14 x 10^12 m^2 So, the sail must have an area of approximately 2.14 x 10^12 m^2 when it absorbs all the incident radiation.

b) Sail perfectly reflects all the incident radiation

: Step 2: Calculate radiation pressure when reflected Radiation pressure (P) when reflected can be found using the formula: P = (2*Intensity of sunlight)/(Speed of light*c), where c = 3.00 x 10^8 m/s P = (2*1.40 kW/m^2 * 1,000 W/kW) / (3.00 * 10^8 m/s) P = 9.33 * 10^(-9) N/m^2 Step 3: Calculate sail area required Area = Force / Radiation pressure Area = 10,000 N / 9.33 * 10^(-9) N/m^2 Area ≈ 1.07 x 10^12 m^2 So, the sail must have an area of approximately 1.07 x 10^12 m^2 when it perfectly reflects all the incident radiation.