Q/C A 65.0-kg Boy And His 40.0-kg Sister, Both Wearing Roller Blades, Face Each Other At Rest.
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Introduction
Imagine a sunny afternoon at a bustling skate park. A 65.0-kg boy and his 40.0-kg sister are standing face-to-face, both wearing roller blades. They are initially at rest, engaging in a playful moment before starting to glide across the concrete. This scenario, though simple at first glance, offers a fascinating insight into the principles of physics—specifically, Newton’s laws of motion, conservation of momentum, and the effects of internal forces on a system at rest.
Understanding how these two individuals interact on their roller blades, especially when they push off or shift their weight, reveals the fundamental concepts of internal and external forces, center of mass, and how action and reaction forces operate in real-world situations. This article aims to analyze the physics involved in this scenario comprehensively, making it accessible to students, educators, and physics enthusiasts alike.
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Context and Relevance of the Scenario
The scenario of a boy and his sister standing face-to-face on roller blades is not only relatable but also serves as an excellent practical example for studying mechanics. It demonstrates:
- Conservation of Momentum: How internal forces within a system do not change the total momentum unless external forces act.
- Center of Mass Dynamics: How the position and movement of the system's center of mass influence the motion.
- Force Interaction: How action and reaction forces operate during pushing and balancing on roller blades.
By analyzing this situation, we can better understand how athletes, dancers, and even astronauts manage motion and balance. It also helps clarify common misconceptions about motion, such as the idea that pushing against each other would cause either person to move significantly or remain stationary.
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Fundamental Physics Principles Involved
Newton’s Laws of Motion
- First Law (Inertia): An object at rest stays at rest unless acted upon by an external force.
- Second Law (F = ma): The acceleration of an object depends on the net external force applied and its mass.
- Third Law (Action-Reaction): For every action, there is an equal and opposite reaction.
Conservation of Momentum
In an isolated system with no external forces, the total momentum remains constant. When the boy and sister push against each other, their internal forces change their individual momenta, but the total momentum of the system remains zero if initially at rest.
Center of Mass
The center of mass (COM) of a system is the point where the entire mass can be considered to be concentrated for analyzing translational motion. For the boy-sister system, the COM's position and movement are crucial in understanding their motions after pushing off.
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Analyzing the Scenario: Step-by-Step
Initial Conditions
- Both are facing each other, at rest.
- Mass of boy, \( m_b = 65.0\, \mathrm{kg} \)
- Mass of sister, \( m_s = 40.0\, \mathrm{kg} \)
- Initial velocities: \( v{b,i} = 0 \), \( v{s,i} = 0 \)
Action: The Push-Off
Suppose the boy and sister push against each other with equal and opposite forces. According to Newton’s third law, these forces are equal in magnitude but opposite in direction.
Resulting Motion
- The boy will move backward with velocity \( v_b \).
- The sister will move forward with velocity \( v_s \).
Because no external horizontal forces are acting (assuming negligible friction), their momenta after pushing will satisfy:
\[
mb vb + ms vs = 0
\]
This equation indicates that the total momentum of the system remains zero, consistent with conservation of momentum.
Calculating Final Velocities
From the momentum equation:
\[
mb vb = - ms vs
\]
Assuming the magnitude of the force and the duration of push are such that the impulses are equal:
\[
vb = - \frac{ms}{mb} vs
\]
The velocities are inversely proportional to their masses:
\[
vb = - \frac{40.0\, \mathrm{kg}}{65.0\, \mathrm{kg}} vs \approx -0.615 v_s
\]
The negative sign indicates opposite directions.
Conservation of Center of Mass
The position of the center of mass before and after the push remains the same, assuming no external forces. The initial position:
\[
X{COM,i} = \frac{mb x{b,i} + ms x{s,i}}{mb + m_s}
\]
Since they face each other at rest, the initial positions are arbitrary but fixed, say:
- \( x_{b,i} = 0 \)
- \( x_{s,i} = d \)
The initial COM:
\[
X_{COM,i} = \frac{65.0 \times 0 + 40.0 \times d}{105.0}
\]
After pushing, their positions change:
- \( xb' = xb + v_b t \)
- \( xs' = xs + v_s t \)
The center of mass position after the push:
\[
X{COM,f} = \frac{mb xb' + ms xs'}{mb + m_s}
\]
Since no external forces act, \( X{COM,f} = X{COM,i} \). This allows us to analyze their velocities and confirm the conservation of momentum.
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Practical Implications and Real-World Applications
Balance and Motion Control in Sports
Understanding how mass and force interactions influence motion helps athletes optimize their techniques. For example, roller skaters, ice skaters, and dancers rely on internal forces to initiate movement without external pushes.
Safety Considerations
Knowing that internal pushes cause movement while the system's overall momentum remains conserved can inform safety protocols, especially when children or beginners learn to avoid unintended falls or collisions.
Engineering and Robotics
Designing systems where internal forces generate motion—such as robots or automated vehicles—requires understanding these physics principles to ensure stability and control.
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Additional Factors Affecting Motion
While the idealized analysis assumes negligible friction and perfect force application, real-world factors include:
- Friction: Between roller blades and the ground, which opposes motion.
- Air Resistance: Slight but becomes relevant at higher speeds.
- Distribution of Force: How force is applied during pushing affects acceleration.
- Body Mechanics: Posture and balance influence the efficiency of motion.
In practice, these factors can influence the velocities achieved and the ease of movement after pushing.
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Conclusion
The scenario of a 65.0-kg boy and his 40.0-kg sister facing each other on roller blades exemplifies several foundational physics principles. From the conservation of momentum to the dynamics of internal forces, analyzing their interaction offers valuable insights into motion and balance.
By understanding that internal forces like pushes cause movement within the system but do not alter the total momentum, learners can better grasp how objects and systems behave in real life. Whether in sports, engineering, or everyday activities, these principles underpin much of the physical world around us.
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