Question

Astronauts are playing catch on the International Space Station. One \(55.0-\mathrm{kg}\) astronaut, initially at rest, throws a baseball of mass \(0.145 \mathrm{~kg}\) at a speed of \(31.3 \mathrm{~m} / \mathrm{s}\). At what speed does the astronaut recoil?

Short answer

Answer: The astronaut recoils at a speed of 0.086 m/s.

Step-by-step solution

Understand the conservation of momentum principle

The conservation of momentum states that in an isolated system (no external forces acting), the total momentum before the event is equal to the total momentum after the event. Mathematically, this can be written as: \(m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final\) In this problem, m1 is the mass of the astronaut, m2 is the mass of the baseball, v1_initial is the initial velocity of the astronaut, v2_initial is the initial velocity of the baseball, v1_final is the final velocity of the astronaut, and v2_final is the final velocity of the baseball.

Define initial momentum

Initially, the astronaut and the baseball are at rest, meaning their initial velocities are 0. Therefore, the initial momentum of the system is 0.

Define final momentum

The final momentum of the system is the sum of the momentum of the astronaut and the momentum of the baseball. This can be expressed as: \(m1 * v1_final + m2 * v2_final\) As mentioned earlier, we know the mass of the astronaut (m1 = 55.0 kg) and the mass of the baseball (m2 = 0.145 kg). The baseball's final speed (v2_final) is 31.3 m/s. Furthermore, since the astronaut's final velocity (v1_final) is in the same direction, its value will be negative. Therefore, this can be rewritten as: \(55.0 * (-v1_final) + 0.145 * 31.3\)

Equate initial and final momentum

By conservation of momentum principle, the initial momentum of the system (0) is equal to the final momentum of the system: \(0 = 55.0 * (-v1_final) + 0.145 * 31.3\)

Solve for the astronaut's recoil speed (v1_final)

Now, we need to solve for v1_final to find the astronaut's recoil speed: \(v1_final = \frac{0.145 * 31.3}{55.0}\) \(v1_final = 0.086 \mathrm{~m} / \mathrm{s}\) The astronaut recoils at a speed of 0.086 m/s.

Key concepts

Laws of Physics

At the very foundation of understanding any physical scenario, including the astronauts playing catch, are the laws of physics. These are the rules that govern how objects behave in our universe, from grand astronomical scales to subatomic levels. Certain principles, like Newton's laws of motion, are pivotal for solving problems related to dynamics, which is the study of forces and how they affect the motion of objects.

Within these laws is the principle of conservation of momentum, which is particularly significant in this astronaut scenario. It's the backbone of predicting outcomes in collision and separation events, where no external forces interfere. This principle is fundamental in physics because it provides a reliable prediction of post-event velocities when masses and initial velocities are known. Students must appreciate the reliability and constancy of these laws as they apply uniformly across different scenarios, whether on Earth or in the microgravity environment of the International Space Station.

Momentum in Isolated Systems

Momentum is a measure of the 'quantity of motion' an object possesses and is a central concept in the study of mechanics. It is defined as the product of an object's mass and its velocity. The situation of the astronaut and baseball on the International Space Station presents a classical case of an isolated system, where the only significant forces acting are internal.

Why is this important to understand? The conservation of momentum indicates that in the absence of external forces, the total momentum of a system remains constant. This is not just a whimsical idea but a robust law that applies without exception to isolated systems. In our example, that means the momentum of the astronaut and the baseball before the throw is equal to their combined momentum afterwards. This principle is crucial when calculating unknown variables such as the astronaut's recoil velocity because it gives us a reliable equation to work with: the combined initial and final momentum must be equivalent.

Recoil Velocity

Recoil velocity is the speed at which one object moves backward after it exerts force on another object, a common observation in everyday actions, such as firing a gun or, as in our example, throwing an object in space. When the astronaut throws the baseball, they will move in the opposite direction due to the conservation of momentum. The principle dictates that they will recoil with a velocity that equals the momentum of the baseball divided by the mass of the astronaut.

This backward movement is what we call the recoil velocity, and it's a vector quantity, meaning it has both magnitude and direction. The value of recoil velocity is centrally dependent on the mass of the objects and the velocity with which force is applied. Understanding the intricacies of recoil is pivotal for students not only for solving textbook exercises but also in grasping real-life applications, such as in the mechanics of propulsion in space.