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Chapter 7: Problem 17

An arrow shot into a straw target penetrates a distance that depends on the speed with which it strikes the target. How does the penetration distance change if the arrow's speed is doubled? Be sure to list all the assumptions you make while arriving at your answer.

Short Answer

Expert verified
If the arrow's speed is doubled, considering other variables like the arrow's mass and the conditions remain constant, the penetration distance is likely to quadruple assuming that all the kinetic energy is absorbed for the penetration.

Step by step solution

01

Understand the problem

The goal is to find the relationship between the speed of the arrow and its penetration distance into the straw target. The basic physics principle that underlies this problem is the concept of kinetic energy.
02

Implement the kinetic energy principle

Helpful to understand that the kinetic energy of an object is given by the formula \( \frac{1}{2}mv^2 \), where m is the mass of the object and v is its velocity. When the arrow hits the target, its kinetic energy is transferred to the work done in penetrating the target.
03

Infer the relationship

Given the formula for kinetic energy, it's clear that the energy is proportional to the square of the velocity. Thus, if the velocity (speed) of the arrow is doubled, the kinetic energy would quadruple (since \( (2v)^2 = 4v^2 \)). It is this kinetic energy that is responsible for the penetration of the arrow into the target, hence the penetration distance would also quadruple assuming all other conditions remain the same.
04

Take note of assumptions

While arriving at the answer, it鈥檚 important to acknowledge the assumptions that were made. One was that the arrow's kinetic energy translates directly into penetration distance, despite the fact that some energy may be lost due to external forces or the destruction of the straw target. The other involved keeping the mass of the arrow constant since the kinetic energy also depends on the arrow鈥檚 mass, which might not always be the case in realistic scenarios.

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Key Concepts

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

Penetration Distance
The penetration distance of an arrow shot into a target is how far the arrow enters or travels into the material it strikes. In our case, we're considering a straw target. Several factors can affect this distance. However, a crucial element is the speed of the arrow upon impact. When analyzing this concept, understanding that the arrow's kinetic energy is transferred to the target is essential. This transfer of energy allows the arrow to overcome resistance and penetrate the target.

Here are some fundamental factors affecting penetration distance:
  • Material properties of the target: A denser or more resistant material could absorb more energy, reducing penetration.
  • Arrow's mass: A heavier arrow can carry more momentum and kinetic energy.
  • Speed of the arrow: As seen, if speed doubles, kinetic energy increases fourfold.
Assumptions made include a constant mass and that energy solely contributes to penetration.
Velocity
An arrow鈥檚 velocity, or speed in a particular direction, plays a pivotal role in determining how deeply it penetrates a target. Velocity enters the kinetic energy equation \[ KE = \frac{1}{2}mv^2 \].

Here, the velocity (\( v \)) is squared, making it a very potent factor. This means:
  • Any change in velocity results in a significant impact on kinetic energy.
  • If velocity doubles, the kinetic energy quadruples due to the square relationship.
It's important to remember, velocity alone doesn't dictate penetration. The arrow鈥檚 mass and target composition also affect it.

In simple terms, velocity is significant because it exponentially increases the energy the arrow carries, rather than increasing linearly.
Work-Energy Principle
The work-energy principle is the backbone of this exercise. It tells us how the arrow's energy, as it strikes the target, gets converted into the work needed to penetrate the material. According to this principle, the work done on an object is equal to the change in its kinetic energy.

Here's how it applies:
  • When an arrow hits its target, its kinetic energy does the work to push through the target material.
  • A quadrupled energy (from the doubled speed) means four times the capacity to perform work (penetrate).
However, one must note energy isn't always fully transferred. Some energy may dissipate as heat or into damaging the target, meaning not all energy translates to penetration. This principle crucially assumes that the majority of kinetic energy is used for penetration under ideal conditions.

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