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) Create free-body diagrams with gravitational forces toward each other. (b) Approx. 109.5 days with assumed constant acceleration. (c) Acceleration increases as distance decreases.

Step 1: Construct Free-Body Diagrams

Draw each astronaut as a circle. For each, show a single force vector pointing towards the other astronaut. Label the force vectors as the gravitational force exerted by one astronaut on the other.

Step 2: Calculate the Gravitational Force

The gravitational force between two masses is given by \[ F = G \frac{m_1 m_2}{r^2} \] where \( G = 6.674 \times 10^{-11} \, \mathrm{m^3 \, kg^{-1} \, s^{-2}} \), \( m_1 = 65 \, \mathrm{kg} \), \( m_2 = 72 \, \mathrm{kg} \), and \( r = 20 \, \mathrm{m} \). Substituting these values, calculate \( F \).

Step 3: Calculate Initial Acceleration for Each Astronaut

Using Newton's Second Law \( F = ma \), solve for the acceleration of each astronaut: \( a_1 = \frac{F}{m_1} \) and \( a_2 = \frac{F}{m_2} \). Use the gravitational force calculated in the previous step.

Step 4: Determine Time to Rendezvous with Constant Acceleration

If acceleration is constant and they start from rest, use the equation for constant acceleration: \[ s = \frac{1}{2} a t^2 \] Solve for \( t \) using the relative acceleration (sum of their accelerations) and replacing \( s \) with 20 m. Convert the time from seconds to days.

Step 5: Discuss Constant Acceleration Assumption

In reality, acceleration would not remain constant because as the astronauts get closer, the gravitational force between them increases due to \( F \propto \frac{1}{r^2} \). Hence, the acceleration would actually increase as they approach each other.

Key Concepts

Newton's Second Law

Newton's Second Law of Motion is an essential principle in physics that describes how the motion of an object changes when it is acted upon by a force. The law is usually written as:\[ F = ma \]where:

• \( F \) is the force applied on the object.
• \( m \) is the mass of the object.
• \( a \) is the acceleration of the object.

This means that the force acting on an object equals the mass of the object multiplied by its acceleration. For the astronauts, as they try to rendezvous in space, each one experiences a force due to the mutual gravitational attraction. This force is what causes them to accelerate towards one another.

By resolving Newton's Second Law in this case, we can calculate their accelerations. The force is derived from the gravitational force equation, and once known, it allows us to understand how quickly each astronaut gains speed towards the other. Understanding Newton’s Second Law is crucial for predicting how objects will move due to varying forces, whether on Earth or in space.

astronauts in space

When considering astronauts in space, it's fascinating because they are in a microgravity environment. In space, far from any planet or star, the dominant force between two objects like astronauts is gravity. Despite their relatively small masses compared to planets, astronauts still exert gravitational forces on each other due to mass, no matter how small, always having this property. Here are a few interesting points about astronauts and gravity:

• Gravity acts on all objects with mass, pulling them together; this is referred to as the gravitational force.
• The gravitational force between two astronauts in space, though small, can still result in noticeable movement over time.
• In this scenario, modeling astronauts as uniform spheres helps simplify the calculation of gravitational forces.

When two astronauts start at rest in space, like in this exercise, they will gradually move toward one another as gravity pulls them closer. Their motion becomes much more predictable when understanding the principles laid out by Newton and the characteristics of gravitational forces.

constant acceleration

Constant acceleration is a scenario where an object's velocity changes at a consistent rate over time. Typically, this means that neither the speed nor the direction of velocity changes without a net force acting upon it in variable intensity. However, in this exercise, focusing on constant acceleration makes solving the rendezvous problem easier.Some considerations include:

• When acceleration is constant, the equations of motion become more straightforward to apply, allowing easier calculation of variables like time or distance.
• The astronauts' initial idea to assume constant acceleration leads to calculating how long it would take for them to meet, as the combined accelerations of both offer an average rate of speeding toward each other.
• Using the formula \( s = \frac{1}{2} a t^2 \) allows us to solve for the time \( t \) for them to meet, assuming their acceleration remains consistent.

However, in reality, as the astronauts get closer, the gravitational force increases and thus the acceleration would increase. So, while the approach with constant acceleration is an academic simplification, it provides necessary insights into motion dynamics when predicting time and distance.

free-body diagram

A free-body diagram is a graphical tool used by physicists to show all the forces acting on a particular object. It helps by breaking down the forces to understand how they affect motion. In this problem, creating a free-body diagram for each astronaut is useful to assess the forces acting upon them. Creating a free-body diagram involves:

• Representing each astronaut as a dot or a simple circle.
• Drawing force vectors that point toward the other astronaut, representing the gravitational attraction.
• Labeling these vectors as forces, which are dependent on factors like mass and distance.

Using a free-body diagram here allows us to visually comprehend that each astronaut experiences a force pulling them toward one another. It's a clear representation of Newton's Second Law in action, showing how the gravitational force results in acceleration. Free-body diagrams simplify complex problems by isolating the forces, making calculations and interpretations more manageable.