Question

(a) Neglecting gravitational forces, what force would be required to accelerate a 1200-metric-ton spaceship from rest to one-tenth the speed of light in 3 days? In 2 months? (One metric ton \(=1000 \mathrm{~kg} .)(b)\) Assuming that the engines are shut down when this speed is reached, what would be the time required to complete a 5-light-month journey for each of these two cases? (Use 1 month \(=30\) days.)

Short answer

To accelerate the spaceship to one-tenth the speed of light, the force required would be \(1.385 \times 10^{14} \mathrm{~N}\) over 3 days and \(6.9 \times 10^{12} \mathrm{~N}\) over 2 months. After the engines are shut down, the time required to complete a 5-light-month journey would be approximately 13.7 years for both cases.

Step-by-step solution

Convert Variables to Standard Units

Convert the spaceship's mass from metric-ton to kg: \(1200 \times 1000 = 1200000 \mathrm{~kg}\). Convert the speed of light (\(3 \times 10^8 \mathrm{~m/s}\)) to one-tenth of its value: \(0.1 \times 3 \times 10^8 = 3 \times 10^7 \mathrm{~m/s}\). Convert time from days and months to seconds: \(3 \mathrm{~days}=3 \times 24 \times 60 \times 60=259200 \mathrm{~s}\) and \(2 \mathrm{~months}=2 \times 30 \times 24 \times 60 \times 60=5184000 \mathrm{~s}\).

Calculate Force for Required Acceleration

Use the formula for acceleration: acceleration = final velocity/ time. Then use the force formula: force = mass x acceleration. Substituting the calculated acceleration, for 3 days, the force is \(1200000 \times \frac{3 \times 10^7}{259200} = 1.385 \times 10^{14} \mathrm{~N}\). And for 2 months, the force is \(1200000 \times \frac{3 \times 10^7}{5184000} = 6.9 \times 10^{12} \mathrm{~N}\).

Calculate Time to Complete Journey

Now using the velocity calculated in the first part and the given distance of 5 light-months, calculate the time required for each case. 1 light-month equals to the distance light travels in one month, which is \(3 \times 10^8 \times 30 \times 24 \times 60 \times 60 =2.592 \times 10^{15} \) m. So, 5 light-months is \(5 × 2.592 \times 10^{15} = 1.296 \times 10^{16} m\). Use time=distance/velocity formula. For 3 days case, time = \(1.296 \times 10^{16}/(3 \times 10^7) = 4.32 \times 10^8 \mathrm{~s} \approx 13.7 \mathrm{~years}\). And for 2 months case, time = \(1.296 \times 10^{16}/(3 \times 10^7) = 4.32 \times 10^8 \mathrm{~s} \approx 13.7 \mathrm{~years}\). So the time required to complete the journey is the same in both cases.

Key concepts

Acceleration

Acceleration is the rate at which an object changes its velocity over time. When a spaceship goes from rest to a certain speed, it undergoes acceleration. In the case of the spaceship in the exercise, it needs to achieve an acceleration to reach one-tenth the speed of light. To calculate acceleration, you need to know the final velocity and the time period over which this change occurs. The formula is:\[ a = \frac{\text{final velocity}}{\text{time}} \]For example, if the spaceship needs to reach a velocity of \(3 \times 10^7 \mathrm{~m/s}\) in 259200 seconds (3 days), the acceleration would be:\[ a = \frac{3 \times 10^7}{259200} = 115.74 \mathrm{~m/s^2} \]Knowing the acceleration is crucial for calculating the force needed to propel the spaceship.

Newton's Second Law

Newton's Second Law of Motion explains the relationship between an object's mass, its acceleration, and the force applied to it. It can be summed up by the equation:\[ F = m \cdot a \]Where:

• \( F \) is the force applied to the object (in Newtons, \(N\)).
• \( m \) is the mass of the object (in kilograms, \(kg\)).
• \( a \) is the acceleration (in meters per second squared, \(m/s^2\)).

In the exercise, to find out the force needed to accelerate the 1200-metric-ton spaceship, you must first determine the acceleration (as previously calculated). Then, multiply the mass by this acceleration.For instance, with a mass of 1200000 kg and an acceleration of 115.74 m/s² for the 3-day case, the force is:\[ F = 1200000 \cdot 115.74 = 1.385 \times 10^{14} \mathrm{~N} \]Understanding this law helps predict how the spaceship will respond to different levels of force.

Speed of Light

The speed of light is considered the ultimate speed limit of the universe. It is the speed at which light travels in a vacuum and is exactly \(3 \times 10^8\) meters per second. Because of this incredible speed, distances in space are often measured in light-years or light-months, which indicates how far light can travel in that time. In this exercise, the spaceship aims to reach a speed that is one-tenth of the speed of light, or \(3 \times 10^7\) m/s. This high velocity significantly affects the time it takes to travel long distances. One light-month is the distance light covers in one month, which is roughly \(2.592 \times 10^{15} \) meters. Therefore, a journey of 5 light-months equates to about \(1.296 \times 10^{16}\) meters in distance. Understanding the speed of light allows us to grasp how vast and immense space travel distances are, emphasizing the enormous time it would take to travel between cosmic objects, even at high speeds.