Question

An arrow of mass \(m=88.0 \mathrm{~g}(0.0880 \mathrm{~kg})\) is fired from a bow. The bowstring exerts an average force of \(F=110 . \mathrm{N}\) on the arrow over a distance \(d=78.0 \mathrm{~cm}(0.780 \mathrm{~m}) .\) Calculate the speed of the arrow as it leaves the bow.

Short answer

Answer: The speed of the arrow as it leaves the bow is 44.15 m/s.

Step-by-step solution

Write the Work-Energy Theorem

The work-energy theorem states that the work done on an object is equal to its change in kinetic energy. Mathematically, it can be written as: \(W = ∆K = K_{f} - K_{i}\) Here, \(W\) is the work done, \(K_{f}\) is the final kinetic energy, and \(K_{i}\) is the initial kinetic energy.

Calculate the work done by the force

The work done by a force over a distance can be found using the formula: \(W = F \cdot d\) Here, \(F\) is the average force and \(d\) is the distance. Given \(F = 110\,\text{N}\) and \(d = 0.780\,\text{m}\), we can find the work done as follows: \(W = 110 * 0.780 = 85.8\,\text{J}\)

Set the initial kinetic energy

Since the arrow starts at rest, its initial kinetic energy is 0. Therefore, \(K_{i} = 0\).

Calculate the final kinetic energy

According to the work-energy theorem, the work done on the arrow is equal to its change in kinetic energy. Since the initial kinetic energy is 0, the change in kinetic energy is just the final kinetic energy. Therefore, we have: \(K_{f} = W = 85.8\,\text{J}\)

Calculate the velocity of the arrow using the kinetic energy formula

The kinetic energy of the arrow is given by the formula: \(K_{f} = \frac{1}{2}mv^{2}\) Here, \(m\) is the mass of the arrow, and \(v\) is its velocity. We want to find the velocity, so we will rearrange this equation to solve for \(v\). First, we will multiply both sides by 2: \(2K_{f} = mv^{2}\) Now, we will divide both sides by \(m\) and take the square root of the result to isolate \(v\): \(v = \sqrt{\frac{2K_{f}}{m}}\) Given \(K_{f} = 85.8\,\text{J}\) and \(m = 0.0880\,\text{kg}\), we can find the velocity as follows: \(v = \sqrt{\frac{2 * 85.8}{0.0880}} = \sqrt{\frac{171.6}{0.0880}} = \sqrt{1949.091} = 44.15\,\text{m/s}\) The speed of the arrow as it leaves the bow is \(44.15\,\text{m/s}\).

Key concepts

Work Done by Force

Understanding the concept of 'work done by force' is essential in physics, especially when examining the motion of objects. Work occurs when a force acts upon an object to cause a displacement. But how do we quantify this? The simple equation to calculate work is:

\[ W = F \times d \]

In this formula, \(W\) represents work, \(F\) is the constant force applied, and \(d\) is the displacement in the direction of the force. If the force is in Newtons (N) and the displacement in meters (m), the work done is expressed in joules (J). It's key to remember that only the component of the force that is in the direction of the displacement does work.

• The force must cause a displacement for work to be done.
• Work can be either positive or negative.
• If the direction of the force and the displacement are the same, the work is positive.
• If they are in opposite directions, the work is negative.

In our arrow example, the bowstring exerts a force which does positive work on the arrow, causing it to accelerate and gain kinetic energy.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is a reflection of the work that has been done to accelerate the object to its current speed. Defined by the equation:

\[ K = \frac{1}{2} mv^2 \]

where \(K\) is the kinetic energy, \(m\) is the mass of the object, and \(v\) is the velocity, kinetic energy increases with the square of the speed—this means that even a small increase in speed will result in a large increase in kinetic energy. For our arrow scenario, as it is propelled by the bowstring, it converts the work done by the force into kinetic energy.

• Kinetic energy is always positive as speed squared is always positive.
• The SI unit for kinetic energy is the joule (J), the same as work.

This fundamental concept helps us understand how objects with mass move and interact based on the work done on them. In the arrow's case, its kinetic energy upon leaving the bow can tell us a lot about the force exerted by the bow and the distance over which it acted.

Physics Problem Solving

Problem-solving is a critical skill in physics and understanding how to approach a problem methodically can make all the difference. For physics problems especially, a step-by-step approach helps in breaking down complex concepts into manageable tasks. Here’s a guideline to tackle such problems effectively:

• Identify the relevant concepts involved, such as the work-energy theorem in the arrow example.
• Translate the problem into mathematical terms with the applicable equations.
• Substitute the given values into the equations.
• Carefully perform the mathematical operations to solve for the unknown.
• Assess the solution to check if it is reasonable and consistent with physical principles.

When we applied this structured strategy to calculate the arrow’s speed, we used the work-energy theorem to relate the work done on the arrow to its kinetic energy. With a formula for kinetic energy and the work calculated, we found the velocity. Physics problem solving requires an analytical mindset, and practice often involves working through diverse problems to develop a deep understanding of the concepts in play.