Question

Astronauts are playing baseball on the International Space Station. One astronaut with a mass of \(50.0 \mathrm{~kg}\), initially at rest, hits a baseball with a bat. The baseball was initially moving toward the astronaut at \(35.0 \mathrm{~m} / \mathrm{s}\), and after being hit, travels back in the same direction with a speed of \(45.0 \mathrm{~m} / \mathrm{s} .\) The mass of a baseball is \(0.140 \mathrm{~kg} .\) What is the recoil velocity of the astronaut?

Short answer

Answer: The recoil velocity of the astronaut is approximately -0.224 m/s, moving in the opposite direction of the final velocity of the baseball.

Step-by-step solution

Write down the conservation of linear momentum equation:

The law of conservation of linear momentum states that the total momentum of an isolated system remains constant, provided that no external forces are acting upon it. Mathematically, this can be represented as: \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\) where: - \(m_1\) and \(m_2\) are the masses of the astronaut and the baseball, respectively; - \(u_1\) and \(u_2\) are their initial velocities; - and \(v_1\) and \(v_2\) are their final velocities.

Identify the given information and plug the values into the equation:

We have the following given information: - Mass of the astronaut: \(m_1 = 50.0 \mathrm{~kg}\) - Mass of the baseball: \(m_2 = 0.140 \mathrm{~kg}\) - Initial velocity of the astronaut: \(u_1 = 0 \mathrm{~m/s}\) (initially at rest) - Initial velocity of the baseball: \(u_2 = -35.0 \mathrm{~m/s}\) (moving toward the astronaut) - Final velocity of the baseball: \(v_2 = 45.0 \mathrm{~m/s}\) (travels back in the same direction) Now, let's plug these values into the conservation of linear momentum equation: \(50.0 \mathrm{~kg} \cdot 0 \mathrm{~m/s} - 0.140 \mathrm{~kg} \cdot 35.0 \mathrm{~m/s} = 50.0 \mathrm{~kg} \cdot v_1 + 0.140 \mathrm{~kg} \cdot 45.0 \mathrm{~m/s}\)

Solve for the unknown value (recoil velocity of the astronaut):

Now let's solve the equation for the unknown value \(v_1\) (recoil velocity of the astronaut): \(-4.9 \mathrm{~kg\cdot m/s} = 50.0 \mathrm{~kg} \cdot v_1 + 6.3 \mathrm{~kg\cdot m/s}\) Moving the terms around to isolate \(v_1\): \(v_1 = \dfrac{-4.9 \mathrm{~kg\cdot m/s} - 6.3 \mathrm{~kg\cdot m/s}}{50.0 \mathrm{~kg}}\) \(v_1 = \dfrac{-11.2 \mathrm{~kg\cdot m/s}}{50.0 \mathrm{~kg}}\) \(v_1 = -0.224 \mathrm{~m/s}\) Thus, the recoil velocity of the astronaut is approximately \(-0.224 \mathrm{~m/s}\). This negative sign indicates that the astronaut moves in the opposite direction of the final velocity of the baseball.

Key concepts

Conservation of Linear Momentum

The principle of conservation of linear momentum is a vital concept in physics, particularly when analyzing collisions and interactions between objects. It states that within an isolated system, the total momentum remains constant unless acted upon by external forces. Momentum, a vector quantity, is the product of an object's mass and its velocity, and it has both magnitude and direction.

When applying this principle to a scenario like an astronaut hitting a baseball in space, it is crucial to remember that momentum is conserved in both magnitude and direction. Before and after the collision (in this case, the hit of the baseball), the sum of the momentum of the astronaut and the baseball must be equal. The concept is often represented by the equation:
\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\).

This equation illustrates that the combined momentum of the astronaut and baseball before the hit is equal to their combined momentum afterward. Through this, the problem becomes a matter of algebra, and by substituting known quantities, we can solve for the unknown ones.

Understanding No External Forces

For momentum conservation to hold true, the astronaut and baseball must be an isolated system with no external forces. This is a good assumption in space, where friction and air resistance are negligible.

Recoil Velocity

Recoil velocity is the speed at which an object moves backward after a collision or the release of a projectile. It is a reaction to the conservation of momentum, and its calculation is essential in understanding the dynamics of movements in physics.

For the astronaut hitting a baseball in space, the recoil velocity is the speed at which the astronaut will move backwards after hitting the ball. The magnitude of this velocity depends on both masses and the relative velocities involved in the process. According to Newton's third law, for every action, there is an equal and opposite reaction, so the astronaut, upon striking the ball, will experience a backward force causing the recoil.

Calculating Recoil

To calculate recoil velocity, we use the same momentum conservation formula. After rearranging and solving for the astronaut's final velocity, we get an understanding of recoil. The exercise solution demonstrates this process step by step, leading to the astronaut's recoil velocity. It's pivotal to note that a negative velocity indicates a direction opposite to that of the baseball after being hit.

Isolated System

An isolated system in physics refers to a collection of objects that do not exchange any matter or energy with the surroundings. Such a system is essential when applying the conservation of momentum.

In the context of an astronaut hitting a baseball on the International Space Station, the astronaut-baseball setup can be considered an isolated system because external forces like gravity and air resistance are virtually absent in the microgravity environment of space. This allows us to assume that no external net force acts on the system, making it viable to apply the principle of conservation of momentum.

Importance in Calculations

Recognizing an isolated system is critical in solving physics problems because it simplifies the calculations. Only internal forces act between the components of the system, meaning that energy and momentum are not lost or gained through outside influence. This simplification is why we can solve for unknown variables, such as recoil velocity, with greater ease and why it's a foundational assumption in many physics problems.