Problem 18 - College Physics

Question

Rendezvous in space! 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 \(65 \mathrm{~kg}\) and the other a mass of \(72 \mathrm{~kg},\) and they start from rest \(20.0 \mathrm{~m}\) 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?

Step-by-Step Solution

Answer

(a) Use gravitational formula to find initial accelerations. (b) Calculate time using kinematics; convert to days. (c) Acceleration increases as distance decreases.

Step 1: Understand the problem

The astronauts are trying to meet in space by relying on their gravitational attraction to one another. Their initial separation is 20.0 meters. We must use this information to determine several things: each person's initial acceleration, the time it would take to meet if that acceleration were constant, and whether the acceleration remains constant or not.

Step 2: Calculate gravitational force

The gravitational force between two objects is given by the formula: \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]where \(G\) is the gravitational constant \( 6.67 \times 10^{-11} \, \mathrm{Nm}^2/\mathrm{kg}^2 \), \(m_1 = 65 \, \mathrm{kg}\), \(m_2 = 72 \, \mathrm{kg}\), and \(r = 20 \, \mathrm{m}\). Substituting these values, we find the gravitational force.

Step 3: Find initial acceleration of each astronaut

Each astronaut experiences an acceleration toward the other due to the gravitational force. Use Newton's second law, \( F = m \cdot a \), to calculate the acceleration. For the 65 kg astronaut: \[ a_1 = \frac{F}{65} \] and for the 72 kg astronaut: \[ a_2 = \frac{F}{72} \].

Step 4: Calculate total constant acceleration

Both accelerations are towards each other, so the effective acceleration bringing them together is the sum: \[ a_{total} = a_1 + a_2 \].

Step 5: Determine time using constant acceleration

If the acceleration were constant, use the kinematic equation:\[ s = ut + \frac{1}{2} a_{total} t^2 \]where \(s = 20 \, \mathrm{m}\), and initial velocity \(u = 0\). Solve for time \(t\) taken for the astronauts to meet.

Step 6: Convert time into days

The time \(t\) obtained from the previous step will be in seconds. Convert this value into days by dividing by the number of seconds in a day (86400 seconds/day).

Step 7: Assess constancy of acceleration

The acceleration depends on their relative positions, which changes as they move closer. As distance \(r\) decreases, gravitational force (and acceleration) increases because \(F\) is inversely proportional to \(r^2\). Thus, their acceleration actually increases over time.

Key Concepts

Newton's Laws

Newton's Laws of Motion are fundamental principles that help us understand how objects behave when forces act upon them. When looking at our astronauts' problem, Newton's second law is particularly useful. It states that the force on an object is equal to the mass of the object multiplied by its acceleration, represented as:

• \[ F = m imes a \]

This law helps us determine the acceleration of each astronaut due to the gravitational force between them. By knowing the masses of the astronauts and the gravitational force acting on each, you can calculate how quickly they move towards each other. Newton's second law provides the framework for understanding how the distance between the astronauts affects their subsequent movement. As it turns out, as the force results from their mass and separation, it proves vital in setting up mathematical models for their journey in space.

Gravitational Force

Gravitational Force is the attractive force that pulls objects toward each other due to their mass. Issac Newton's universal law of gravitation allows us to calculate the gravitational force between two masses with the formula:

• \[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

Here, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the objects, and \( r \) is the distance between their centers. This force is incredibly small for human-sized masses at typical distances, like between our two astronauts at 20 meters, yet it's still significant in the vacuum of space without other forces acting. However, as they move closer (reducing \( r \)), the force increases, pulling them progressively faster towards each other. This principle explains why gravitational force grows stronger as the astronauts drift nearer, contrary to the thought of constant acceleration.

Acceleration

Acceleration is the rate of change of velocity of an object as a result of a force applied over time. It can be calculated using Newton's second law, where the force and mass determine the acceleration experienced. For our astronauts, we found the acceleration by dividing the gravitational force by each astronaut's mass:

• \[ a_1 = \frac{F}{m_1} \]
• \[ a_2 = \frac{F}{m_2} \]

These formulas allow each astronaut's initial acceleration toward the other to be calculated, leading to their eventual meeting in space. Since both accelerations are directed towards each other, they effectively sum up to bring the astronauts together faster. However, it is critical to note that this acceleration isn't constant over time due to the dependence on distance \( r \), which is constantly decreasing as they get closer to each other.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. To solve how the astronauts move toward each other, kinematic equations come into play, in particular:

• \[ s = ut + \frac{1}{2} a_{total} t^2 \]

This equation helps determine how long it takes for the astronauts to rendezvous given their initial conditions (starting from rest) and the assumption of constant acceleration. Here, \( s \) represents the separation distance, \( u \) the initial velocity (zero in this case), \( a_{total} \) the sum of accelerations of both astronauts, and \( t \) the time it takes to meet. Solving this gives a time value in seconds, which can further be converted to days to find out how long they would have to wait if the conditions remained unchanged. However, considering the variable nature of gravitational pull, the real scenario is different as the acceleration increases as they move towards each other, shortening the time predicted by this simple model.