7. Schwab fires a handgun straight upwards into the air (very foolishly). If the bullet leaves the muzzle of the gun with a velocity of 240m/s and a kinetic energy of 284 J, how high will the bullet travel?

Here's how to solve this problem:<br /><br />**1. Understand the Concepts**<br /><br />* **Conservation of Energy:** As the bullet travels upwards, its kinetic energy is converted into potential energy. At the highest point, all kinetic energy becomes potential energy.<br />* **Kinetic Energy:** KE = (1/2)mv²<br />* **Potential Energy:** PE = mgh<br />* **Gravity:** The acceleration due to gravity (g) is approximately 9.8 m/s² downwards.<br /><br />**2. Find the Mass of the Bullet**<br /><br />We are given the initial kinetic energy and velocity. We can use the kinetic energy formula to find the mass (m) of the bullet:<br /><br />KE = (1/2)mv²<br />28.1 J = (1/2)m * (240 m/s)²<br />56.2 J = m * 57600 m²/s²<br />m = 56.2 J / 57600 m²/s²<br />m ≈ 0.000976 kg (approximately 1 gram, which is reasonable for a bullet)<br /><br />**3. Calculate the Maximum Height**<br /><br />At the highest point, all the initial kinetic energy is converted to potential energy:<br /><br />KE_initial = PE_final<br />(1/2)mv² = mgh<br /><br />Notice that the mass (m) cancels out on both sides:<br /><br />(1/2)v² = gh<br /><br />Now, solve for h (height):<br /><br />h = v² / (2g)<br />h = (240 m/s)² / (2 * 9.8 m/s²)<br />h = 57600 m²/s² / 19.6 m/s²<br />h ≈ 2939 m<br /><br />**Answer:** The bullet will travel approximately 2939 meters high.<br /><br /><br />**Important Note:** This calculation ignores air resistance. In reality, air resistance would significantly reduce the maximum height reached by the bullet.<br />