Ch. 09 - Linear Momentum

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 09 - Linear Momentum

Problem 15

Chapter 9, Problem 15

A mass mₐ = 2.0 kg, moving with velocity $\vec{v_{a}}$ = (4.0 î + 5.0 ĵ ― 2.0 k̂) m/s, collides with mass m₈ = 3.0 kg, which is initially at rest. Immediately after the collision, mass mₐ is observed traveling at velocity $\vec{v_{a}^{'}}$ = (― 2.0 î + 3.0 k̂) m/s. Find the velocity of mass m₈ after the collision. Assume no outside force acts on the two masses during the collision.

Verified step by step guidance

Step 1: Understand the principle of conservation of momentum. In the absence of external forces, the total momentum of a system before and after a collision remains constant. Mathematically, this is expressed as:

m ₐ v ₐ + m ₈ v ₈ = m ₐ v'ₐ + m ₈ v'₈

, where

v ₈

is the initial velocity of mass

m ₈

(which is zero in this case), and

v'₈

is the velocity of mass

m ₈

after the collision.

Step 2: Write the momentum conservation equation for each component (x, y, and z) separately. Since momentum is a vector quantity, we must conserve momentum in all three directions. For the x-component:

m ₐ v ₐₓ = m ₐ v'ₐₓ + m ₈ v'₈ₓ

. For the y-component:

m ₐ v ₐᵧ = m ₐ v'ₐᵧ + m ₈ v'₈ᵧ

. For the z-component:

m ₐ v ₐz = m ₐ v'ₐz + m ₈ v'₈z

Step 3: Substitute the given values into the equations. For the x-component:

2.0 (4.0) = 2.0 (- 2.0) + 3.0 v'₈ₓ

. For the y-component:

2.0 (5.0) = 2.0 (0) + 3.0 v'₈ᵧ

. For the z-component:

2.0 (- 2.0) = 2.0 (3.0) + 3.0 v'₈z

Step 4: Solve each equation for the components of

v'₈

. For the x-component, isolate

v'₈ₓ

v'₈ₓ = (

expression after simplification

)

. Repeat for the y- and z-components.

Step 5: Combine the components to express the final velocity of mass

m ₈

as a vector:

v'₈ = v'₈ₓ î + v'₈ᵧ ĵ + v'₈z k̂

. Substitute the calculated values for

v'₈ₓ

v'₈ᵧ

, and

v'₈z

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system with no external forces, the total momentum before a collision is equal to the total momentum after the collision. Momentum is a vector quantity, calculated as the product of mass and velocity. In this scenario, the momentum of both masses before and after the collision must be equated to find the unknown velocity of mass m₈.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this problem, the velocities of the masses are represented as vectors, and their components must be added separately in the x, y, and z directions. Understanding how to manipulate these vectors is crucial for accurately calculating the final velocity of mass m₈ after the collision.

Elastic vs. Inelastic Collisions

Collisions can be classified as elastic or inelastic based on whether kinetic energy is conserved. In this case, since the problem does not specify that kinetic energy is conserved, it is likely an inelastic collision, where the two masses may stick together or simply exchange momentum. Recognizing the type of collision helps in applying the correct principles to solve for the unknowns in the problem.

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