Astronauts are playing catch on the International Space Station. One 55.0 -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?

Momentum is quite literally the 'oomph' of motion. It is a measurement that combines an object’s mass and velocity to demonstrate the object’s capacity to affect other objects with its motion. Mathematically, momentum (\textbf{p}) is expressed as the product of an object's mass (\textbf{m}) and its velocity (\textbf{v}):
\( p = mv \).
Importantly, momentum is a vector quantity, which means it has both magnitude and direction, thus the velocity's direction is as crucial as the speed itself in defining momentum. In any event involving the interaction of objects, such as collision or separation, the total momentum of the system remains consistent before and after the event if no external forces are acting upon it. This principle is famously known as the conservation of momentum.

Recoil velocity is the speed at which an object moves back after exerting a force on another object. It's best visualized when considering interactions where two objects interact and then move apart, such as a gun firing a bullet or, in our astronaut's case, throwing a baseball. Due to the conservation of momentum, whatever impulse is given to the baseball, an equal and opposite impulse is experienced by the astronaut, making the astronaut recoil.
The magnitude of the recoil velocity can be less apparent than that of the thrown object because the mass of the recoiling object is typically much greater, as shown in our exercise. We used
\( v_a' = \frac{-m_b v_b'}{m_a} \)
to calculate the astronaut's recoil velocity, confirming that the astronaut moves back much slower than the baseball is thrown forward.

Sir Isaac Newton fundamentally changed our understanding of physics with his three laws of motion. The third law is often succinctly stated as 'For every action, there is an equal and opposite reaction.' This means that forces always come in pairs; if object A exerts a force on object B, object B simultaneously exerts a force of equal magnitude and opposite direction on object A. This is the underlying reason why the astronaut in our problem recoils when throwing the baseball. The force applied to the baseball to propel it forward results in an equal force pushing the astronaut backward. Newton's Third Law is crucial for understanding why the total momentum remains unchanged in an isolated system, even after objects within it interact.

An isolated system in physics is idealized as a collection of objects upon which no external forces act. Inside such a system, physical quantities like momentum are conserved. For our astronaut and baseball, space acts as an isolated system—if we neglect forces such as gravity from nearby masses and resistive forces (which are minimal in space)—leading to an ideal demonstration of conservation of momentum. In our problem, the combined momentum of the astronaut and baseball before the throw is exactly equal to the combined momentum afterward. Since they started at rest, their total initial momentum was zero, and after the throw, their momentum had to balance out to maintain a total of zero due to the lack of external forces acting on the system.