A spacecraft performs a burn to increase its velocity by 350 m/s. If its initial speed was 7,200 m/s, what is its new kinetic energy per kg? Use \( KE = \frac{1}{2} v^2 \).

["## Boosting Spacecraft Velocity: Calculating New Kinetic Energy After a Burn", "When a spacecraft performs a propulsion burn, its velocity increases, directly impacting its kinetic energy. Understanding how velocity changes translate into energy gains is essential in orbital mechanics and mission planning. In this article, we explore a specific example: a spacecraft whose initial speed is 7,200 m/s increases its velocity by 350 m/s through a burn. We calculate its new kinetic energy per kilogram using the fundamental kinetic energy formula.", "## Understanding the Basics: Kinetic Energy Formula", "Kinetic energy (( KE )) is the energy an object possesses due to its motion. For a spacecraft moving at velocity ( v ), the kinetic energy per unit mass is given by:", "[\nKE = \frac{1}{2} v^2\n]", "This formula assumes the spacecraft’s mass is effectively 1 kg for simplicity in energy calculations—why per kg matters: expressing energy per mass standardizes comparisons between different spacecraft, regardless of their size, and simplifies performance analysis.", "## Step-by-Step: From Initial Velocity to New Kinetic Energy", "### Step 1: Determine the new velocity", "Initial speed:\n[\nv_{\ ext{initial}} = 7,!200 \ ext{ m/s}\n]", "Velocity increase:\n[\n\Delta v = 350 \ ext{ m/s}\n]", "New speed:\n[\nv_{\ ext{new}} = v_{\ ext{initial}} + \Delta v = 7,!200 + 350 = 7,!550 \ ext{ m/s}\n]", "### Step 2: Apply the kinetic energy formula", "With ( v_{\ ext{new}} = 7,!550 ) m/s:\n[\nKE = \frac{1}{2} \ imes (7,!550)^2\n]", "Compute ( 7,!550^2 ):\n[\n7,!550^2 = 57,!025,!000 \ ext{ m}^2/\ ext{s}^2\n]", "Now divide by 2:\n[\nKE = \frac{1}{2} \ imes 57,!025,!000 = 28,!512,!500 \ ext{ J/kg}\n]", "## Final Result: New Kinetic Energy Per kg", "The spacecraft’s new kinetic energy per kilogram is:\n[\n\boxed{28,!512,!500 \ ext{ Joules per kilogram}}\n]", "### Why This Matters in Space Missions", "Even small velocity changes significantly affect kinetic energy—a critical factor in orbital transfers, escape maneuvers, and fuel efficiency. This example concretely demonstrates how propulsion burns translate into measurable energy gains, emphasizing the precision required in spacecraft design and mission trajectory planning.", "For aerospace engineers, mastering these calculations ensures optimal burn durations and fuel use, maximizing mission success and efficiency in the unforgiving environment of space.", "---", "Keywords: spacecraft kinetic energy, velocity burn calculation, KE = 1/2 v², orbital mechanics, space mission energy, per kilogram kinetic energy, propulsion burn effects"]