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Ch. 09 - Linear Momentum
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 09 - Linear MomentumProblema 96
Capitolo 9, Problema 96

A fake hockey puck of mass 4m has been rigged to explode. Initially the puck is at rest on a frictionless ice rink. Then it bursts into three pieces. One chunk, of mass m, slides across the ice at velocity vî. Another chunk, of mass 2m, slides across the ice at velocity 2v ĵ. Determine the velocity of the third chunk.

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Start by applying the principle of conservation of momentum. Since the puck was initially at rest, the total momentum of the system before the explosion is zero. Therefore, the total momentum of the three pieces after the explosion must also sum to zero.
Write the momentum equation in vector form. Let the velocity of the third chunk (mass m) be \( \vec{v}_3 = v_{3x} \hat{i} + v_{3y} \hat{j} \). The total momentum equation becomes: \( m \vec{v}_1 + 2m \vec{v}_2 + m \vec{v}_3 = 0 \), where \( \vec{v}_1 = v \hat{i} \) and \( \vec{v}_2 = 2v \hat{j} \).
Separate the momentum equation into components. For the x-direction: \( m(v) + 0 + m(v_{3x}) = 0 \). For the y-direction: \( 0 + 2m(2v) + m(v_{3y}) = 0 \).
Solve for \( v_{3x} \) in the x-direction equation: \( v_{3x} = -v \). Then solve for \( v_{3y} \) in the y-direction equation: \( v_{3y} = -4v \).
Combine the components to express the velocity of the third chunk as a vector: \( \vec{v}_3 = -v \hat{i} - 4v \hat{j} \).

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Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In this scenario, since the puck is initially at rest, the total momentum before the explosion is zero. After the explosion, the vector sum of the momenta of all pieces must also equal zero, allowing us to solve for the unknown velocity of the third chunk.
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Conservation Of Momentum

Vector Addition

Vector addition is the process of combining vectors to determine a resultant vector. In this problem, the velocities of the chunks are represented as vectors, and their components must be added separately in the x and y directions. This allows for the calculation of the third chunk's velocity by ensuring that the total momentum in each direction sums to zero.
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Vector Addition By Components

Mass and Velocity Relationship

The relationship between mass and velocity is crucial in momentum calculations, as momentum is defined as the product of mass and velocity (p = mv). In this case, the masses of the chunks and their respective velocities will be used to express the momentum of each piece. Understanding how these quantities interact is essential for applying the conservation of momentum to find the unknown velocity.
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Relationships Between Force, Field, Energy, Potential
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