/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 83 A \(4.00-\mathrm{g}\) bullet, tr... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 83

A \(4.00-\mathrm{g}\) bullet, traveling horizontally with a velocity of magnitude 400 \(\mathrm{m} / \mathrm{s}\) , is fired into a wooden block with mass 0.800 \(\mathrm{kg}\) , initially at rest on a level surface. The bullet passes through the block and emerges with its speed reduced to 120 \(\mathrm{m} / \mathrm{s} .\) The block shides a distance of 45.0 \(\mathrm{cm}\) along the surface from its initial position. (a) What is the coefficient of kinetic friction between block and surface? (b) What is the decrease in kinetic energy of the bullet? (c) What is the kinetic energy of the block at the instant after the bullet passes through it?

Short Answer

Expert verified
(a) Coefficient of friction is 0.223. (b) Decrease in bullet's kinetic energy is 291.2 J. (c) Block's kinetic energy is 0.784 J.

Step by step solution

01

Calculate the initial kinetic energy of the bullet

The initial kinetic energy (KE) of the bullet can be calculated using the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the bullet and \( v \) is its velocity. Given that the mass of the bullet is \( 4.00 \, \mathrm{g} = 0.004 \, \mathrm{kg} \) and its initial velocity is \( 400 \, \mathrm{m/s} \), the initial kinetic energy can be calculated as follows:\[KE_i = \frac{1}{2} \times 0.004 \, \mathrm{kg} \times (400 \, \mathrm{m/s})^2 = 320 \, \mathrm{J}\]
02

Calculate the final kinetic energy of the bullet

The final kinetic energy of the bullet after it has passed through the block can be calculated similarly using its final speed, \( 120 \, \mathrm{m/s} \):\[KE_f = \frac{1}{2} \times 0.004 \, \mathrm{kg} \times (120 \, \mathrm{m/s})^2 = 28.8 \, \mathrm{J}\]
03

Calculate the decrease in kinetic energy of the bullet

The decrease in kinetic energy of the bullet is the difference between its initial and final kinetic energies:\[\Delta KE = KE_i - KE_f = 320 \, \mathrm{J} - 28.8 \, \mathrm{J} = 291.2 \, \mathrm{J}\]
04

Determine the velocity of the block after the bullet passes through

To find the block's velocity immediately after the bullet passes through, we apply the principle of conservation of momentum. Initial momentum is from the bullet only, while final momentum accounts for both bullet and block:\[0.004 \, \mathrm{kg} \times 400 \, \mathrm{m/s} = 0.004 \, \mathrm{kg} \times 120 \, \mathrm{m/s} + 0.8 \, \mathrm{kg} \times v_{block}\]Solve for \( v_{block} \). After simplifying:\[v_{block} = \frac{0.004 \times (400 - 120)}{0.8} = 1.4 \, \mathrm{m/s}\]
05

Calculate the kinetic energy of the block

The kinetic energy of the block right after the bullet passes through is calculated using the velocity from Step 4:\[KE_{block} = \frac{1}{2} \times 0.8 \, \mathrm{kg} \times (1.4 \, \mathrm{m/s})^2 = 0.784 \, \mathrm{J}\]
06

Calculate the work done by kinetic friction

The work done by kinetic friction \( W_f \) is equal to the kinetic energy lost, which is also equal to the force of friction multiplied by the distance the block slides. Friction force \( f \) can be expressed as \( f = \mu_k \cdot N \), where \( \mu_k \) is the coefficient of friction and \( N \) is the normal force. The distance \( d \) is 0.45 m. Using:\[W_f = f \cdot d = \mu_k \cdot m_{block} \cdot g \cdot d\]Equating the work done to the kinetic energy of the block from Step 5, solve for \( \mu_k \):\[0.784 \, \mathrm{J} = \mu_k \times 0.8 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} \times 0.45 \, \mathrm{m}\]\[\mu_k = \frac{0.784}{0.8 \times 9.8 \times 0.45} \approx 0.223\]

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Key Concepts

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

Kinetic Energy
Kinetic energy (KE) is the energy an object possesses due to its motion. It's a crucial concept in physics, particularly when analyzing scenarios involving moving objects. The formula for kinetic energy is given by \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object, and \( v \) is its velocity. This formula tells us that kinetic energy is directly proportional to the mass of the object and the square of its velocity.

In the exercise involving the bullet and the wooden block, we first calculate the kinetic energy of the bullet when it is fired with a velocity of 400 m/s. Then, we compute its kinetic energy again after it exits the block at 120 m/s. The decrease in kinetic energy signifies the amount of energy transferred or lost during this interaction, mainly due to internal forces like friction and deformation.

This change in kinetic energy of the bullet is also crucial in determining how much energy is available to do work in moving the block. Understanding these concepts will help us delve deeper into the principles of energy conservation and transfer during collisions or interactions between objects.
Coefficient of Friction
The coefficient of friction, particularly the kinetic coefficient, represents the relationship between the force necessary to keep two surfaces sliding past each other and the normal force pressing them together. It's a dimensionless value and is crucial in predicting the movement of objects across surfaces. The formula is given by \( f = \mu_k \cdot N \), where \( f \) is the frictional force, \( \mu_k \) is the coefficient of kinetic friction, and \( N \) is the normal force.

In the exercise, we determine the coefficient of kinetic friction between the block and the surface it slides on. After the bullet passes through the block, the remaining kinetic energy causes the block to slide. By knowing how far it slides and using the work-energy principle, we can calculate the frictional force acting on the block. This force is then used to solve for \( \mu_k \).

By equating the work done by the friction force to the kinetic energy lost due to friction, the coefficient of kinetic friction is found. This value helps understand the interaction between the block and the surface, giving insights into the resistive forces at play.
Energy Loss
Energy loss is a common occurrence in physical interactions, especially when kinetic energy is transferred between objects or converted into other forms of energy like heat or sound due to friction. In analyzing energy loss, it's essential to account for where the energy goes and how it affects the system's dynamics.

In the bullet and block scenario, the energy loss primarily happens when the bullet passes through the block. The initial kinetic energy of the bullet decreases as some of its energy transfers to the block, allowing it to move. This transfer is not 100% efficient, as evidenced by the decrease in the bullet's kinetic energy.

Additionally, some kinetic energy converts into thermal energy due to friction between the block and the surface as it slides. Energy loss can therefore be seen in both the block and the bullet, affecting their subsequent motions. Understanding how energy is lost and where it goes is key to comprehending dynamic systems and energy conservation principles in physics.

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Most popular questions from this chapter

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