91Ó°ÊÓ

Two girls (masses \(m_{1}\) and \(m_{2}\) ) are on roller skates and stand at rest, close to each other and face to face. Girl-1 pushes squarely against girl-2 and sends her moving backward. Assuming the girls move freely on their skates, write an expression for the speed with which girl-1 moves. We take the two girls to comprise the system under consideration. The problem states that girl-2 moves "backward," so let that be the negative direction; therefore, the "forward" direction is positive. There is no resultant external force on the system (the push of one girl on the other is an internal force), and so momentum is conserved: 1 from which Girl-1 recoils with this speed. Notice that if \(m_{2} / m_{1}\) is very large, \(v_{1}\) is much larger than \(v_{2}\). The velocity of girl-1, \(\overrightarrow{\mathbf{v}}_{1}\), points in the positive forward direction. The velocity of girl-2, \(\overrightarrow{\mathbf{v}}_{2}\), points in the negative backward direction. If we put numbers into the equation, \(\mathrm{v}_{2}\) would have to be negative and \(u_{1}\) would come out positive.

Step by step solution

01

Understanding the Systems and Forces

We begin by considering the system comprised of two girls, where both girls are initially at rest. The push by girl-1 on girl-2 is an internal force, meaning there is no external force acting on the system, so momentum is conserved. Girl-2 moves in the negative direction (backward) and girl-1 in the positive direction (forward).

02

Expression for Conservation of Momentum

By the principle of conservation of momentum, the total momentum before the push equals the total momentum after the push. Since both girls start at rest, the total initial momentum of the system is zero: \[ 0 = m_1 v_1 + m_2 v_2 \]where \( v_1 \) and \( v_2 \) are the velocities of girl-1 and girl-2, respectively.

03

Rearrange the Equation

Rearrange the equation for conservation of momentum to solve for the velocity of girl-1:\[ m_1 v_1 = -m_2 v_2 \]We multiply \( v_2 \) by negative because it is in the opposite direction of \( v_1 \).

04

Solve for the Speed of Girl-1

To find the expression of the speed of girl-1 (\( v_1 \)), divide both sides by \( m_1 \):\[ v_1 = -\frac{m_2}{m_1} v_2 \]This equation gives us the velocity of girl-1 as a function of girl-2's velocity and the ratio of their masses.

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Key Concepts

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

Newton's Third Law

Newton's Third Law states that for every action, there's an equal and opposite reaction. This law is crucial in understanding the interaction between the two girls on roller skates. When girl-1 pushes girl-2, girl-2 pushes back on girl-1 with an equal but opposite force. This is what makes girl-1 move backward after the push.

Physical interactions like this rely not just on forces, but also on the way these forces act. In this case, the internal push by girl-1 serves as the action and the push back by girl-2 as the reaction. This means that each girl experiences the other's force, helping us describe their subsequent motion.

In practical terms, Newton's Third Law is the reason both girls move upon the push, and why the momentum of the system—comprising both girls—remains balanced.

Internal Forces

Internal forces occur between components within a system, like in our example with the two girls. The push that girl-1 applies to girl-2, also called a mutual force, doesn't affect the system's overall external momentum since it's an internal interaction.

Internal forces are vital in closed systems like this because they allow us to maintain a consistent analysis of the system’s momentum. Here, the absence of external forces ensures momentum conservation, so any change in motion by one girl is intrinsically linked to the change in motion of the other.

For the problem involving the girls, understanding internal forces helps us appreciate why they can move freely on their skates without any external influence affecting the sum of their movements.

System of Particles

In the context of the problem, the two girls on skates represent a simple system of particles. A system of particles is simply a group of objects that we study together to analyze their collective interactions and subsequent motions.

The concept emphasizes that when considering interactions within this system, like the push between the girls, we mostly deal with internal forces. Since these systems can incorporate complex interactions, defining a clear system helps isolate specific behaviors and avoid confusion.

By treating the girls as a system, it becomes clearer how they collectively maintain the same total momentum they had initially (which was zero), irrespective of how they move relative to one another within the system.

Momentum Calculation

Momentum is a key quantity when analyzing moving systems, defined as the product of mass and velocity (\(p = mv\)). In our exercise, we initially set the system's momentum to zero because neither girl is moving before the push.

Following the conservation of momentum principle, the total momentum before and after the push remains constant, equating to zero. This principle is articulated mathematically as follows:\[0 = m_1 v_1 + m_2 v_2\]

From here, solving for girl-1's speed involves isolating the variable, leading to this expression:\[v_1 = -\frac{m_2}{m_1} v_2\]

This calculation underlines the inversely proportional relationship between the velocities based on their mass ratio, thereby capturing the essence of how each girl’s motion influences the other in terms of speed and direction.