Question

The Sun yacht Diana, designed to navigate in the solar system using the pressure of sunlight, has a sail area of \(3.1 \mathrm{~km}^{2}\) and a mass of \(930 \mathrm{~kg}\). Near Earth's orbit, the Sun could exert a radiation force of \(29 \mathrm{~N}\) on its sail. (a) What acceleration would such a force impart to the craft? (b) A small acceleration can produce large effects if it acts steadily for a long enough time. Starting from rest then, how far would the craft have moved after 1 day under these conditions? ( \(c\) ) What would then be its speed? (See "The Wind from the Sun," a fascinating science fiction account by Arthur C. Clarke of a Sun yacht race.)

Short answer

The acceleration the sunlight's radiation force imparts to the yacht is found by dividing the force by the yacht's mass, the distance traveled by the yacht after a day is calculated using the relevant equation of motion, and the final speed of the yacht after a day is determined using the formula \( v = u + at \). The detailed calculations will provide the exact numerical values.

Step-by-step solution

Calculation of acceleration using Newton's second law

According to Newton's second law, the force \( F \) on an object is equal to its mass \( m \) multiplied by its acceleration \( a \). So, the acceleration of the yacht can be calculated using the equation: \( a = F/m \)Substituting the provided values:\( a = 29\, \mathrm{N} / 930\, \mathrm{kg} \)

Calculation of distance travelled using equations of motion

The equation of motion used to calculate the distance \( s \) travelled by the yacht when it starts from rest is: \( s = ut + 0.5at^2 \)where\( u \) is the initial speed, which is 0 since the yacht is at rest,\( a \) is the acceleration calculated in Step 1,\( t \) is the time, which is one day. Convert this time into seconds that is \( t = 1 * 24 * 3600\, \mathrm{s} \)

Calculation of speed using equations of motion

The final velocity or speed \( v \) of the yacht can be calculated using formula:\( v = u + at \)Substitute the values:\( u = 0 \) (yacht starts from rest),\( a \) is from Step 1, and \( t \) is from Step 2.

Key concepts

Understanding Newton's Second Law

An essential principle in physics is Newton's second law of motion, which states that the acceleration of an object is proportional to the net force acting upon it and inversely proportional to its mass. In symbolic terms, this law is often written as \( F = ma \), where \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is the resulting acceleration.

For instance, in the context of the solar sail exercise, the yacht Diana experiences a force due to light photons from the sun colliding with its enormous sail. By applying the law \( a = F/m \) and inserting the provided values, we easily calculate the yacht's acceleration. The elegance of Newton's second law lies in its simplicity and universal application across various scenarios, from a solar sail gliding through space to a car accelerating on the highway.

Deciphering Equations of Motion

Equations of motion allow us to predict the future position and velocity of an object moving under constant acceleration. Specifically, the equation \( s = ut + 0.5at^2 \) helps us find the distance \( s \) traveled by an object with initial velocity \( u \) after time \( t \) with constant acceleration \( a \). In the case of the Sun yacht Diana, with an initial velocity of zero, this equation simplifies and becomes a powerful tool to forecast the yacht’s journey through space.

This calculation is particularly helpful for understanding long-term movement in space, where even small, consistent forces like radiation pressure can lead to significant displacement over time, highlighting the cumulative power of steady forces.

Radiation Pressure and Solar Sailing

Radiation pressure is the force exerted by light when it strikes a surface. Although light has no mass, it has momentum, which can be transferred to objects, like a sail, upon collision. In the case of the solar sail, this principle enables propulsion without fuel; light from the Sun exerts pressure on the sail, causing the craft to accelerate.

Understanding radiation pressure is key to utilizing solar sails for space exploration. The Sun yacht Diana serves as a perfect example: the exercise demonstrates how radiation pressure translates into a calculable force, which we can then use to determine the yacht’s acceleration and subsequent displacement over time.

These calculations are not just theoretical; they are practical considerations for future space missions that might harness the power of sunlight to travel across the solar system. As continuous propulsion systems, solar sails present a sustainable and cost-effective method for long-distance space travel.