Solar wind (radiation) that is incident on the top of Earth's atmosphere has an average intensity of \(1.3 \mathrm{kW} / \mathrm{m}^{2} .\) Suppose that you are building a solar sail that is to propel a small toy spaceship with a mass of \(0.1 \mathrm{kg}\) in the space between the International Space Station and the moon. The sail is made from a very light material, which perfectly reflects the incident radiation. To assess whether such a project is feasible, answer the following questions, assuming that radiation photons are incident only in normal direction to the sail reflecting surface. (a) What is the radiation pressure (force per \(\mathrm{m}^{2}\) ) of the radiation falling on the mirror-like sail? (b) Given the radiation pressure computed in (a), what will be the acceleration of the spaceship when the sail has of an area of \(10.0 \mathrm{m}^{2}\) ? (c) Given the acceleration estimate in (b), how fast will the spaceship be moving after 24 hours when it starts from rest?

Step by step solution

01

Find the radiation pressure

First, we need to find the radiation pressure on the solar sail. The radiation pressure is given by the formula \(P = \frac{I}{c}\), where \(I\) is the intensity of the radiation and \(c\) is the speed of light. Use \(I = 1.3kW/ m^{2}\) and \(c = 3 * 10^{8}m/s\): \(P = \frac{1.3 * 10^{3} W / m^{2}}{3 * 10^{8}m/s}\)

02

Calculate the force on the solar sail

Using the radiation pressure calculated in Step 1, we can find the force \(F\) on the solar sail by multiplying the pressure \(P\) with the sail's area \(A\): \(F = P * A\)

03

Calculate the acceleration of the spaceship

Use Newton's second law to calculate the acceleration of the toy spaceship. Newton's second law states that \(F = m*a\), where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. Rearrange the equation to solve for the acceleration: \(a = \frac{F}{m}\)

04

Calculate the velocity after 24 hours

The spaceship starts from rest. Using constant acceleration's formula \(v = at\), calculate the spaceship's velocity \(v\) after 24 hours (86400 seconds), where \(a\) is the acceleration and \(t\) is the time: \(v = a * 86400\) Now, we have the step-by-step solution to the given exercise.

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