Although they don’t have mass, photons—traveling at the speed of light—have momentum. Space travel experts have thought of capitalizing on this fact by constructing solar sails—large sheets of material that would work by reflecting photons. Since the momentum of the photon would be reversed, an impulse would be exerted on it by the solar sail, and—by Newton’s Third Law—an impulse would also be exerted on the sail, providing a force. In space near the Earth, about 3.84·1021 photons are incident per square meter per second. On average, the momentum of each photon is 1.30·10–27 kg m/s. Assume that we have a 1291 kg spaceship starting from rest attached to a square sail 27.9 m wide.
a) How fast could the ship be moving after 45 days? b) How many months would it take the ship to attain a speed of 7.75 km/s, roughly the speed of the space shuttle in orbit?
To solve this problem, we can use the concept of conservation of momentum. The momentum change of the photons will be equal to the momentum change of the spaceship, and we can calculate the resulting velocity.
a) To calculate the ship's velocity after 45 days, we need to find the total impulse exerted on the sail during this time.
First, let's calculate the number of photons incident on the sail per square meter per second:
Number of photons = 3.84 x 10^21 photons/m^2/s
Now, we can calculate the momentum change of each photon:
Momentum change = 2 x (momentum of each photon)
= 2 x (1.30 x 10^(-27) kg m/s)
= 2.60 x 10^(-27) kg m/s
Next, we need to find the area of the sail:
Area of sail = width x width
= 27.9 m x 27.9 m
= 777.21 m^2
Since each incident photon exerts an impulse on the sail, the total impulse exerted on the sail over 45 days can be calculated as follows:
Total impulse = Number of photons x Momentum change x Area of sail x Time
= (3.84 x 10^21 photons/m^2/s) x (2.60 x 10^(-27) kg m/s) x (777.21 m^2) x (45 days x 24 hours/day x 60 minutes/hour x 60 seconds/minute)
Now, convert the time to seconds:
Total impulse = (3.84 x 10^21 photons/m^2/s) x (2.60 x 10^(-27) kg m/s) x (777.21 m^2) x (45 days x 24 hours/day x 60 minutes/hour x 60 seconds/minute)
Finally, we can calculate the ship's velocity by dividing the total impulse by the mass of the ship:
Ship's velocity = Total impulse / Mass of the ship
= Total impulse / 1291 kg
b) To calculate how many months it would take for the ship to reach a velocity of 7.75 km/s, we can use a similar approach.
First, convert the velocity to meters per second:
Velocity = 7.75 km/s
= 7.75 x 10^3 m/s
Then, set up the equation to find the time required:
Ship's velocity = Total impulse / Mass of the ship
7.75 x 10^3 m/s = Total impulse / 1291 kg
Solve for the total impulse:
Total impulse = Ship's velocity x Mass of the ship
Now, calculate the time required in seconds:
Time required = Total impulse / (Number of photons x Momentum change x Area of sail)
= (Ship's velocity x Mass of the ship) / (3.84 x 10^21 photons/m^2/s x 2.60 x 10^(-27) kg m/s x 777.21 m^2)
Finally, convert the time to months:
Time required = Time required in seconds / (number of seconds in a minute x number of minutes in an hour x number of hours in a day x number of days in a month)
To calculate the speed of the spaceship after a certain time period using a solar sail, we need to consider the momentum gained from the photons reflected by the sail.
a) To find the speed of the ship after 45 days:
First, we need to calculate the total impulse exerted on the sail by the incident photons.
Impulse (I) = Change in momentum
The momentum change of a single photon can be calculated using the given momentum value:
Momentum change per photon = 2 * (momentum of each photon) = 2 * (1.30 x 10^(-27) kg m/s)
Now, we need to find the number of photons incident on the sail in 45 days.
Number of photons = (photons per square meter per second) * (area of the sail) * (time in seconds)
Area of the sail = (27.9 m) * (27.9 m) = 778.41 m^2
Time in seconds = (45 days) * (24 hours per day) * (60 minutes per hour) * (60 seconds per minute)
Substituting the values into the equation:
Number of photons = (3.84 x 10^21 photons/m^2/s) * (778.41 m^2) * (45 days * 24 * 60 * 60 s/day)
Now, we can calculate the total impulse exerted on the sail:
Total impulse = (Number of photons) * (Momentum change per photon)
Next, we can calculate the speed gained by the spaceship using Newton's second law, which relates force, mass, and acceleration:
Force = Total impulse / (time of flight)
Mass = 1291 kg
Time of flight = 45 days * 24 * 60 * 60 seconds
Acceleration = Force / Mass
Finally, we can calculate the speed gained by the spaceship using the equation:
Speed = Initial speed + (acceleration * time)
Given that the spaceship starts from rest, the initial speed is 0.
b) To calculate the time it takes for the spaceship to reach a speed of 7.75 km/s:
We need to reverse the process and find the time taken using the given speed and the acceleration calculated from the solar sail.
Rearranging the equation: Speed = Initial speed + (acceleration * time)
We can solve for time by substituting the known values:
Time = (Speed - Initial speed) / acceleration
Substituting the values and calculating the time in seconds, we can convert it into months. Since the speed is given in km/s, we need to convert it to m/s for consistency.
Assuming 1 month has 30 days, we can calculate the number of months needed.