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 85c

Chapter 9, Problem 85c

A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collides head-on with an 8.0-kg object moving in the ―𝓍 direction at 4.0 m/s. Determine the final velocity of each object if the 5.5-kg object is at rest after the collision.

Verified step by step guidance

Step 1: Start by identifying the type of collision. Since the problem specifies that the 5.5-kg object is at rest after the collision, this is an inelastic collision where one object comes to rest. The principle of conservation of momentum will be used to solve the problem.

Step 2: Write the equation for the conservation of linear momentum. The total momentum before the collision equals the total momentum after the collision. Mathematically, this is expressed as:

m

₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f, where

m

₁ and

m

₂ are the masses of the two objects,

v

_1i and

v

_2i are their initial velocities, and

v

_1f and

v

_2f are their final velocities.

Step 3: Substitute the known values into the momentum conservation equation. The given values are:

m

₁ = 5.5 \, \(\text{kg}\),

m

₂ = 8.0 \, \(\text{kg}\),

v

_1i = 6.5 \, \(\text{m/s}\),

v

_2i = -4.0 \, \(\text{m/s}\), and

v

_1f = 0. The equation becomes:

(5.5)(6.5) + (8.0)(-4.0) = (5.5)(0) + (8.0)v

_2f.

Step 4: Simplify the equation to solve for

v

_2f. Perform the arithmetic operations on the left-hand side to find the total initial momentum, then divide by the mass of the second object to isolate

v

_2f. The equation simplifies to:

v

_2f = \(\frac{(5.5)(6.5) + (8.0)(-4.0)}{8.0}\).

Step 5: Interpret the result. The value of

v

_2f will indicate the final velocity of the 8.0-kg object after the collision. Ensure the sign of the result is consistent with the direction of motion (positive for +𝓍 direction, negative for ―𝓍 direction).

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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 the total momentum of a closed system remains constant if no external forces act on it. In collisions, the momentum before the collision equals the momentum after the collision. This concept is crucial for analyzing the velocities of objects involved in a collision.

Momentum Calculation

Momentum is calculated as the product of an object's mass and its velocity (p = mv). In this scenario, we need to calculate the initial momentum of both objects before the collision and set it equal to the final momentum after the collision, considering the given conditions of the problem.

Elastic vs. Inelastic Collisions

Collisions can be classified as elastic or inelastic based on whether kinetic energy is conserved. In this case, since the 5.5-kg object comes to rest after the collision, it indicates an inelastic collision where kinetic energy is not conserved, and some energy is transformed into other forms, such as heat or sound.

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Related Practice

Textbook Question

A gun fires a bullet vertically into a 1.40-kg block of wood at rest on a thin horizontal sheet, Fig. 9–54. If the bullet has a mass of 15.0 g and a speed of 230 m/s, how high will the block rise into the air after the bullet becomes embedded in it?

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Textbook Question

Astronomers estimate that a 2.0-km-diameter asteroid collides with the Earth once every million years. The collision could pose a threat to life on Earth. Assume a spherical asteroid has a mass of 3200 kg for each cubic meter of volume and moves toward the Earth at 15 km/s. How much destructive energy could be released when it embeds itself in the Earth?

Textbook Question

A rifle is aimed at a 2.0-kg block of wood along an inclined plane making an angle of 25°, as shown in Fig. 9–59. A 9.5-g bullet is fired at 760 m/s and becomes embedded in the block. How far up the incline does the block/bullet slide?

(a) Ignore the friction.

(b) Assume μₖ = 0.33.

Textbook Question

An astronaut of mass 210 kg including his suit and jet pack wants to acquire a velocity of 2.0 m/s to move back toward his space shuttle. Assuming the jet pack can eject gas with a velocity of 35 m/s, what mass of gas will need to be ejected?

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Textbook Question

A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collides head-on with an 8.0-kg object moving in the ―𝓍 direction at 4.0 m/s. Determine the final velocity of each object if the collision is elastic.

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Textbook Question

A 5.5-kg object moving in the +𝓍 direction at 6.5 m/s collides head-on with an 8.0-kg object moving in the ―𝓍 direction at 4.0 m/s. Determine the final velocity of each object if the objects stick together.

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