Question

An arrow of mass \(m=88 \mathrm{~g}(0.088 \mathrm{~kg})\) is fired from a bow. The bowstring exerts an average force of \(F=110 \mathrm{~N}\) on the arrow over a distance \(d=78 \mathrm{~cm}(0.78 \mathrm{~m})\) Calculate the speed of the arrow as it leaves the bow.

Short answer

Answer: The final speed of the arrow is 44.1 m/s.

Step-by-step solution

Calculate the work done by the force

We can calculate the work done (W) using the formula \(W = F \cdot d \cdot \cos(\theta)\), where F is the force exerted, d is the distance, and θ is the angle between the force and the distance. In this case, the force and distance are in the same direction, so cos(θ) is equal to 1. Therefore, the equation simplifies to \( W = F \cdot d\). Given F = 110 N and d = 0.78 m, we can calculate the work done: \(W = 110 \mathrm{~N} \cdot 0.78 \mathrm{~m} = 85.8 \mathrm{~J}\)

Calculate the change in kinetic energy

Since the arrow starts at rest, the initial kinetic energy is 0. The work done by the force equals the change in kinetic energy, so the final kinetic energy (KE) of the arrow is 85.8 J.

Use the kinetic energy formula to find the final speed

The formula for the kinetic energy of an object is \(KE = \frac{1}{2}mv^2\), where m is the mass of the object and v is its velocity. We can use this formula to find the final speed (v) of the arrow: \(85.8 \mathrm{~J} = \frac{1}{2}(0.088 \mathrm{~kg})v^2\) First, we need to isolate the velocity squared (v^2) on one side of the equation: \(v^2 = \frac{85.8 \mathrm{~J}}{0.5(0.088 \mathrm{~kg})} = 1945.45\) Now, we can take the square root to find the final speed of the arrow: \(v = \sqrt{1945.45} = 44.1 \mathrm{~m/s}\) The final speed of the arrow is 44.1 m/s as it leaves the bow.

Key concepts

Work-Energy Principle

The work-energy principle is pivotal in understanding how objects move and how their motion relates to energy transfer. In simple terms, it states that if work is done on an object, this results in a change in the object's kinetic energy. In other words, work can be seen as the transfer of energy to or from an object.

In the case of the arrow being fired from a bow, we see this principle plain as day. The force of the bowstring does work on the arrow, which effectively transfers energy into the motion of the arrow. Since the arrow starts from rest, all of the work done on it directly converts into its kinetic energy. Understanding this principle helps students grasp how forces acting over distances can impact an object's speed and energy.

Kinetic Energy Formula

When delving into the world of physics, we often discuss 'kinetic energy,' the energy of motion. The kinetic energy formula is expressed as \( KE = \frac{1}{2}mv^2 \), where \( m \) stands for mass and \( v \) stands for velocity. This fundamental formula encapsulates how the speed and mass of an object contribute to its kinetic energy.

For example, knowing the mass of an arrow and the work done on it by the bow, we can rearrange the kinetic energy formula to solve for the arrow's velocity when it leaves the bow. In physics problems, it's common to manipulate such formulas to solve for an unknown variable based on the information provided.

Force and Motion

The relationship between force and motion is a cornerstone of physics, rooted in Newton's laws of motion. In essence, an unbalanced force acting on an object causes it to accelerate. The size of this acceleration depends on both the mass of the object and the magnitude of the force applied.

Through this lens, we consider the arrow: the bowstring's force propels it forward. It's this force that sets the arrow in motion, overcoming the inertia that kept it at rest. The motion of the arrow—a combination of its velocity and trajectory—is the result of this applied force, demonstrating how forces directly influence the motion of objects in our everyday experiences and intricate physics scenarios alike.