Solving the Mystery of Potential Energy and Kinetic Energy in Projectile Motion

How do potential and kinetic energy relate to each other in projectile motion?

When a crossbow shoots a 1.0-kg arrow, it gives it a kinetic energy of 450 J. How much potential energy will the arrow have at the top of its path if the crossbow shoots it straight up into the air?

Understanding the Relationship Between Potential and Kinetic Energy in Projectile Motion

When dealing with projectile motion, potential and kinetic energy are key concepts that play a crucial role in understanding the behavior of the object in motion. The relationship between potential and kinetic energy can be explained through the conservation of mechanical energy.

When an object is launched into the air, like the arrow from the crossbow in this scenario, it possesses kinetic energy due to its motion. As the object moves against the force of gravity, its kinetic energy gradually transforms into potential energy, reaching a maximum at the highest point of its path.

Using the principle of conservation of mechanical energy, we can conclude that the potential energy of the arrow at the top of its path is equal to the kinetic energy it had when it was initially fired from the crossbow. This relationship showcases the transfer between potential and kinetic energy throughout the projectile motion.

Solving the Problem

Let's dive deeper into the specific example provided in the question. We are given that the arrow, with a mass of 1.0 kg, has a kinetic energy of 450 J when fired from the crossbow. The task is to determine the potential energy of the arrow at the top of its path.

By applying the conservation of mechanical energy, we find that the potential energy at the highest point is equivalent to the initial kinetic energy of the arrow. Utilizing the provided information, we can calculate the velocity of the arrow and then determine its potential energy.

Step-by-Step Solution

To solve this problem, we start by understanding the initial kinetic energy of the arrow, which is given as 450 J. By setting this as the kinetic energy of the arrow at the highest point, we can calculate the velocity of the arrow using the formula for kinetic energy.

After determining the velocity, we can then find the potential energy of the arrow at the top of its path by equating it to the calculated kinetic energy. This step-by-step approach enables us to unravel the mystery of potential and kinetic energy transfer in projectile motion.