Question

An arrow is placed on a bow, the bowstring is pulled back, and the arrow is shot straight up into the air; the arrow then comes back down and sticks into the ground. Describe all of the changes in work and energy that occur.

Short answer

energy remains constant until the arrow reaches the ground. #Short_Answer# As the arrow is placed on the bow and the bowstring is pulled back, an increase in potential energy occurs due to the work done by the person pulling the bowstring. When the bowstring is released, this potential energy is converted into kinetic energy, propelling the arrow upwards. At its peak height, the arrow's kinetic energy is fully converted into gravitational potential energy. As the arrow falls back to the ground, its potential energy is converted back into kinetic energy, with the sum of the kinetic and potential energy remaining constant throughout the ascent and descent.

Step-by-step solution

Arrow placed on the bow and bowstring pulled back

When the arrow is placed on the bow and the bowstring is pulled back, the potential energy of the bow and arrow system increases. The increase in potential energy comes from the work done by the person pulling the bowstring back, which involves a force being exerted over a distance.

Arrow shot into the air

As the bowstring is released, the potential energy stored in the bow is converted into kinetic energy, propelling the arrow upwards. The change in energy is given by the work-energy principle, where the work done by the bowstring and the change in potential energy would equal the change in kinetic energy.

Arrow reaches peak height

Once the arrow reaches its peak height, its kinetic energy is fully converted into gravitational potential energy (relative to the ground). During this phase as the arrow ascends, the work done by gravity is negative as it opposes its motion. The sum of the kinetic energy and potential energy remains constant.

Arrow falls back to the ground

As the arrow starts to fall back to the ground, its gravitational potential energy is converted back into kinetic energy. Gravity does positive work during this phase since it now works in the same direction as the motion of the arrow. The sum of the kinetic energy and potential

Key concepts

Conservation of Energy

The concept of the conservation of energy is a fundamental principle in physics which asserts that energy cannot be created or destroyed, only transformed from one form to another. This means that the total amount of energy in a closed system remains constant over time.

In the context of our archery example, when the arrow is shot into the air, the energy in the system is converted but the total amount stays the same. Initially, as the bowstring is pulled back, the energy of the system is stored as elastic potential energy. As the arrow is released and moves upwards, this elastic potential energy is transformed into kinetic energy, due to the force exerted by the released bowstring. At the arrow's peak, all the kinetic energy has been converted into gravitational potential energy. As the arrow falls, this potential energy changes back into kinetic energy until the arrow hits the ground. Throughout this entire process, the total energy - the combination of kinetic and potential - remains unchanged, illustrating the conservation of energy principle.

Potential Energy

Potential energy represents the stored energy in an object or system due to its position or arrangement. There are various forms of potential energy, but in the scenario of an arrow shot from a bow, we deal with two types: elastic potential energy and gravitational potential energy.

Elastic potential energy is stored when the bow is drawn. The force used to pull back the bowstring distorts the bow, storing energy in the process. This energy is what propels the arrow forward when the bowstring is released. On the other hand, gravitational potential energy comes into play as the arrow ascends. The arrow gains height and, with respect to the Earth, the potential energy increases due to the gravitational pull. At the arrow's maximum height, all of its initial kinetic energy has been converted into gravitational potential energy.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It's given by the equation \( KE = \frac{1}{2}mv^2 \), where \(m\) is the mass of the object and \(v\) is its velocity. Whenever an object speeds up or slows down, its kinetic energy changes.

In our exercise, the kinetic energy is at its maximum when the arrow is released from the bow and is moving at its highest speed. As the arrow ascends, air resistance and gravity slow it down, causing a loss in kinetic energy, which is transferred to potential energy. When the arrow pauses momentarily at the peak of its flight, the kinetic energy is zero. As gravity accelerates the arrow back to the earth, the kinetic energy increases again, as the gravitational potential energy decreases correspondingly.

Work-Energy Principle

The work-energy principle links the concepts of work and energy, stating that work done on an object results in a change in its kinetic energy. Work is defined as the exertion of a force over a distance, and it's considered positive when the direction of the force and the direction of movement align.

Applying this principle to the arrow example demonstrates how the work done by the archer's hand in drawing the bow is transferred into the potential energy of the bow and arrow system. When the arrow is released and flies into the air, the work previously done is now manifested as kinetic energy. Likewise, as the arrow reaches its maximum height, the negative work done by gravity (acting against the direction of motion) has gradually reduced the arrow's kinetic energy to zero, converting it all into potential energy. The return trip to the ground involves gravity doing positive work, which converts potential energy back into kinetic energy.