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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?

Short Answer

Expert verified
Answer: The astronaut recoils with a speed of 0.0825 m/s.

Step by step solution

01

Identify the masses and velocities of both bodies before and after the event

Let \(m_a\) be the mass of the astronaut, \(m_b\) be the mass of the baseball, \(v_a\) be the velocity of the astronaut, and \(v_b\) be the velocity of the baseball. From the problem, we know that \(m_a = 55.0 \mathrm{~kg}\), \(m_b = 0.145 \mathrm{~kg}\), and \(v_b = 31.3 \mathrm{~m/s}\). Initially, everything is at rest, so \(v_a = 0\) and \(v_b = 0\) before the event. We want to find \(v_a'\), the velocity of the astronaut after throwing the baseball.
02

Use the conservation of momentum principle to set up an equation

The conservation of momentum principle states that the total momentum before the event is equal to the total momentum after the event. Mathematically, this is expressed as: $$(m_a v_a + m_b v_b) = (m_a v_a' + m_b v_b')$$ Since initially everything is at rest, \(v_a = 0\) and \(v_b = 0\), so the total momentum before the event is 0. We can rearrange the equation to find the velocity of the astronaut after the event: $$v_a' = \frac{-m_b v_b'}{m_a}$$
03

Plug in the given values and solve for the unknown variable

Now, we plug in the given values for \(m_a\), \(m_b\), and \(v_b'\): $$v_a' = \frac{- (0.145 \mathrm{~kg})(31.3 \mathrm{~m/s})}{55.0 \mathrm{~kg}}$$ Simplify the equation to obtain the velocity of the astronaut: $$v_a' = \frac{-4.5375 \mathrm{~kg m/s}}{55. 0 \mathrm{~kg}}$$ $$v_a' = -0.0825 \mathrm{~m/s}$$
04

Interpret the result

The astronaut recoils with a speed of \(0.0825 \mathrm{~m/s}\) in the opposite direction of the baseball's motion, as indicated by the negative sign.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum
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
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.
Newton's Third Law
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.
Isolated Systems
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.

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Most popular questions from this chapter

A bored boy shoots a soft pellet from an air gun at a piece of cheese with mass \(0.25 \mathrm{~kg}\) that sits, keeping cool for dinner guests, on a block of ice. On one particular shot, his 1.2-g pellet gets stuck in the cheese, causing it to slide \(25 \mathrm{~cm}\) before coming to a stop. According to the package the gun came in, the muzzle velocity is \(65 \mathrm{~m} / \mathrm{s}\). What is the coefficient of friction between the cheese and the ice?

Many nuclear collisions are truly elastic. If a proton with kinetic energy \(E_{0}\) collides elastically with another proton at rest and travels at an angle of \(25^{\circ}\) with respect to its initial path, what is its energy after the collision with respect to its original energy? What is the final energy of the proton that was originally at rest?

How fast would a \(5.00-\mathrm{g}\) fly have to be traveling to slow a \(1900 .-\mathrm{kg}\) car traveling at \(55.0 \mathrm{mph}\) by \(5.00 \mathrm{mph}\) if the fly hit the car in a totally inelastic head-on collision?

In waterskiing, a "garage sale" occurs when a skier loses control and falls and waterskis fly in different directions. In one particular incident, a novice skier was skimming across the surface of the water at \(22.0 \mathrm{~m} / \mathrm{s}\) when he lost control. One ski, with a mass of \(1.50 \mathrm{~kg},\) flew off at an angle of \(12.0^{\circ}\) to the left of the initial direction of the skier with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\). The other identical ski flew from the crash at an angle of \(5.00^{\circ}\) to the right with a speed of \(21.0 \mathrm{~m} / \mathrm{s} .\) What was the velocity of the \(61.0-\mathrm{kg}\) skier? Give a speed and a direction relative to the initial velocity vector.

Bats are extremely adept at catching insects in midair. If a 50.0-g bat flying in one direction at \(8.00 \mathrm{~m} / \mathrm{s}\) catches a \(5.00-\mathrm{g}\) insect flying in the opposite direction at \(6.00 \mathrm{~m} / \mathrm{s}\), what is the speed of the bat immediately after catching the insect?

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