Problem 9 - College Physics

Question

Recoil speed of the earth. In principle, any time someone jumps up, the earth moves in the opposite direction. To see why we are unaware of this motion, calculate the recoil speed of the earth when a 75 kg person jumps upward at a speed of 2.0 \(\mathrm{m} / \mathrm{s} .\) Consult Appendix E as needed.

Step-by-Step Solution

Answer

The Earth's recoil speed is approximately \(-2.51 \times 10^{-23}\, \mathrm{m/s}.\)

Step 1: Understand the Concept of Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In this case, the system is the person and the Earth. When the person jumps, the momentum they gain upward is equal and opposite to the momentum the Earth gains downward.

Step 2: Write the Momentum Conservation Equation

Let the person's mass be \(m_p = 75\, \text{kg}\) and the Earth's mass be \(m_e = 5.972 \times 10^{24}\, \text{kg}\). The initial momentum is zero since neither the person nor the Earth is moving. When the person jumps, let \(v_p = 2.0\, \text{m/s}\) be the velocity of the person. The conservation of momentum gives us: \[ m_p \times v_p + m_e \times v_e = 0 \] where \(v_e\) is the Earth's recoil speed.

Step 3: Solve for Earth's Recoil Speed

Rearrange the momentum equation to solve for \(v_e\): \[ v_e = -\frac{m_p \times v_p}{m_e} \]Substitute the given values into the equation:\[ v_e = -\frac{75 \times 2.0}{5.972 \times 10^{24}} \]Perform the calculations to find \(v_e\).

Step 4: Calculate and Interpret the Result

Calculate the recoil speed:\[ v_e = -\frac{150}{5.972 \times 10^{24}} \approx -2.51 \times 10^{-23} \text{ m/s} \]The negative sign indicates that the Earth's movement is in the opposite direction of the person's jump. This minuscule speed explains why this motion is unnoticeable.

Key Concepts

Recoil Speed

When we hear the term "recoil speed," it often brings to mind the motion that occurs when an object moves in response to another force, like when a person jumps. In the case of a jump, the Earth experiences a teeny tiny recoil speed, moving in the opposite direction to the person's jump. This happens because the person and Earth form a pair reacting to mutual forces. However, due to the enormous difference in their masses, with Earth's mass being exceptionally large, the resulting motion of Earth is exceedingly small. This tiny speed, calculated to be approximately -2.51 x 10^-23 m/s, is practically imperceptible, highlighting the enormity of Earth's mass compared to a human. Recoil speed captures this phenomenon, demonstrating the transfer of momentum between interacting objects without necessarily altering the larger body's apparent state.

Momentum Equation

The momentum equation is pivotal when discussing conservation principles in physics. It conveys how momentum before and after an event remains unchanged in a closed system devoid of external forces. In simpler terms, the total momentum of interacting objects remains constant. This relationship is mathematically represented by the equation:\[ m_p \times v_p + m_e \times v_e = 0 \]Here \(m_p\), \(v_p\), \(m_e\), and \(v_e\) denote the masses and velocities of the person and Earth, respectively. The equation shows that any gain in momentum by the jumping person (\(m_p \times v_p\)) is equally counterbalanced by Earth's opposite momentum (\(m_e \times v_e\)). Solving this equation gives us the measure of Earth's recoil speed, reinforcing the concept of equilibrium in motion dynamics.

Closed System

A closed system is a fascinating concept in physics that involves a defined environment where external factors are negligible or removed entirely. When dealing with momentum and its conservation, a closed system ensures that only internal forces are considered, ignoring external influences. This idea is essential to accurately applying the law of conservation of momentum. For instance, when a person jumps, the interaction between the person and the Earth is perfectly encapsulated as a closed system. No outside forces such as wind or friction interfered, thus allowing us to accurately predict motion outcomes following the principles of conservation. This assumption simplifies the problem, ensuring we focus precisely on the effects of internal forces — the jump in this case and Earth's reaction. Understanding closed systems helps us model real-world scenarios in simplified terms, keeping the core physical interactions at the forefront.

Conservation Laws

Conservation laws in physics are foundational principles that maintain certain quantities constant throughout any process, unless influenced by external factors. Among these, the conservation of momentum stands out as a critical tool for analyzing object interactions. This principle asserts that total momentum remains consistent when dealing with a closed system. In our example, the jump of a 75 kg individual at a speed of 2.0 m/s creates a system where their upward momentum is counterbalanced by Earth's downward, yet minuscule, momentum, so overall system momentum doesn't change. Conservation laws help us understand and predict how objects will behave when forces act upon them. They offer a lens into the intrinsic balance present in physical systems, illustrating how individual dynamics feed into larger processes seamlessly. Mastery of these laws provides a deeper comprehension of motion and equilibrium in our daily interactions and the universe at large.