Question

A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. One of them has a mass of 65 kg and the other a mass of 72 kg, and they start from rest 20.0 m apart. (a) Make a free- body diagram of each astronaut, and use it to find his or her initial acceleration. As a rough approximation, we can model the astronauts as uniform spheres. (b) If the astronauts' acceleration remained constant, how many days would they have to wait before reaching each other? (Careful! They \(both\) have acceleration toward each other.) (c) Would their acceleration, in fact, remain constant? If not, would it increase or decrease? Why?

Short answer

(a) Initial accelerations are \(1.20 \times 10^{-11}\, \text{m/s}^2\) and \(1.08 \times 10^{-11}\, \text{m/s}^2\); (b) If constant, \(\approx 15.28\) days to meet; (c) Acceleration increases as they approach.

Step-by-step solution

Calculate Gravitational Force

First, we use Newton's law of universal gravitation to calculate the force of attraction between the two astronauts. The formula is\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \] where \( G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \) (gravitational constant),\( m_1 = 65 \, \text{kg} \), \( m_2 = 72 \, \text{kg} \), and \( r = 20.0 \, \text{m} \). Substitute the values:\[ F = \frac{6.674 \times 10^{-11} \cdot 65 \cdot 72}{(20)^2} = 7.8 \times 10^{-10} \, \text{N} \]

Determine Initial Acceleration

Use Newton’s second law to find the acceleration of each astronaut. For astronaut 1 with mass \( m_1 = 65 \, \text{kg} \), \[ a_1 = \frac{F}{m_1} = \frac{7.8 \times 10^{-10}}{65} \approx 1.20 \times 10^{-11} \, \text{m/s}^2 \]For astronaut 2 with mass \( m_2 = 72 \, \text{kg} \), \[ a_2 = \frac{F}{m_2} = \frac{7.8 \times 10^{-10}}{72} \approx 1.08 \times 10^{-11} \, \text{m/s}^2 \].

Time Calculation (Theoretical Constant Acceleration)

Calculate the time it would take for them to meet if their accelerations were constant. The sum of accelerations is:\[ a_{\text{total}} = a_1 + a_2 = \approx 2.28 \times 10^{-11} \, \text{m/s}^2 \]Using the equation \( r = \frac{1}{2} a_{\text{total}} t^2 \), solve for \( t \):\[ t = \sqrt{\frac{2r}{a_{\text{total}}}} = \sqrt{\frac{2 \cdot 20}{2.28 \times 10^{-11}}} \approx 1.32 \times 10^6 \, \text{s} \]Convert this to days: \( \frac{1.32 \times 10^6}{86400} \approx 15.28 \text{ days} \).

Assessing Constant Acceleration Assumption

Discuss whether the acceleration remains constant. As the astronauts get closer, the distance \( r \) decreases, causing the gravitational force, and thus the acceleration, to increase according to the formula:\[ a = \frac{G \cdot m_1}{r^2} \text{ for astronaut 2, and vice versa} \]Thus, the acceleration does not remain constant; it increases as they approach each other.

Key concepts

Newton's law of universal gravitation

Newton's law of universal gravitation is a fundamental concept describing how every pair of masses exerts an attractive force on each other. The strength of this force is determined by the masses involved and the distance between them. By using the formula \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), we can calculate the gravitational force between two objects. Here, \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2) \), \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of the two masses.

In the exercise, astronauts use each other's gravitational pull to draw themselves closer. Despite the vastness of space, even small gravitational interactions like this have real effects due to the law's universal nature. This calculation forms the foundation for understanding how they will approach each other without any other forces involved.

Free-body diagram

A free-body diagram is a visual tool used in physics to explore all the forces acting on an object. For this exercise, each astronaut can be modeled as a point mass, simplifying their bodies into spheres where gravitational forces can be applied uniformly. In a free-body diagram:

• The gravitational force exerted on the astronaut by the other is depicted by an arrow pointing towards the other astronaut's mass.
• There might be additional forces, but in this scenario, gravity is the only considered force, so the diagram will mainly focus on this.

The diagram helps in visualizing that each astronaut not only experiences gravitational attraction from the other but also exerts it themselves, thus allowing the calculation of individual accelerations based on force interactions.

Acceleration

When discussing acceleration with regard to these astronauts, we're determining how quickly their velocities change due to gravitational force. Acceleration in this context is calculated using Newton's second law. Upon applying the gravitational force formula and obtaining a force value, we can determine how that force causes a change in speed over time for each astronaut. This can be done using \( a = \frac{F}{m} \) for each astronaut.

For instance, as calculated in the exercise, the smaller mass experiences a slightly higher acceleration due to the same force applied over a lesser mass, and vice versa. Acceleration shows us how the astronauts’ speeds change, making them gradually move toward each other. It’s important to remember that at all times, both astronauts accelerate toward each other, contributing to a total acceleration value which is the sum of their individual accelerations.

Newton's second law

Newton's second law of motion is crucial in understanding the movement of the astronauts in this exercise. This law states that the acceleration of an object depends on the net force acting upon it and the object's mass, expressed in the formula: \( F = m \cdot a \). It underpins how we can believably predict each astronaut's initial acceleration responding to gravitational force.

In the example, the force calculated using Newton's law of universal gravitation was used here to find each astronaut's acceleration. This law provides a direct relationship between force and motion; when a force is applied, there is a resulting change in movement depending on the mass. Therefore, when both astronauts start accelerating toward each other, it is this principle that allows calculation of how quickly they will close the gap. Understanding Newton's second law also hints that as they close in, their acceleration increases due to increasing gravitational force, reinforcing that acceleration isn’t constant but dynamic as they converge.