A rocket accelerates from rest at 12 m/s² for 30 seconds. What is its final velocity and the distance traveled during this time?
["Rocket Acceleration: Calculating Final Velocity and Distance Traveled in 30 Seconds", "When a rocket accelerates from rest, understanding its motion is fundamental in physics and space engineering. This article explores a classic scenario: a rocket accelerating at 12 m/s² for 30 seconds, starting from rest. We’ll calculate the final velocity and the distance traveled during this time, a perfect example of uniformly accelerated motion.", "### Situation Overview", "The rocket begins motion at rest, meaning its initial velocity ( u = 0 , \ ext{m/s} ). It accelerates uniformly at a rate of ( a = 12 , \ ext{m/s}^2 ) over a time interval of ( t = 30 , \ ext{seconds} ). To determine its final velocity and the distance it travels, we use the core equations of motion.", "---", "### Final Velocity of the Rocket", "For uniformly accelerated motion, the final velocity ( v ) is given by:", "[\nv = u + a t\n]", "Substitute the known values:", "[\nv = 0 + (12 , \ ext{m/s}^2)(30 , \ ext{s}) = 360 , \ ext{m/s}\n]", "Final Velocity: The rocket reaches a speed of 360 meters per second after 30 seconds.", "---", "### Distance Traveled by the Rocket", "To find the distance traveled, use the equation:", "[\ns = u t + \frac{1}{2} a t^2\n]", "Since ( u = 0 ), the formula simplifies to:", "[\ns = \frac{1}{2} a t^2\n]", "Substitute the values:", "[\ns = \frac{1}{2} (12 , \ ext{m/s}^2)(30 , \ ext{s})^2 = \frac{1}{2} \ imes 12 \ imes 900 = 5400 , \ ext{meters}\n]", "Distance Traveled: The rocket covers 5400 meters during this acceleration phase.", "---", "### Summary", "- Final velocity: 360 m/s\n- Distance traveled: 5400 meters\n- Acceleration: 12 m/s² for 30 seconds from rest", "Understanding these calculations helps engineers plan rocket trajectories, predict performance, and ensure safe and efficient space missions. Whether launching satellites or exploring distant planets, the principles of uniformly accelerated motion form the foundation of rocket propulsion analysis.", "---", "### Key Takeaway", "For rocket acceleration from rest at constant acceleration ( a ) over time ( t ):", "- Final velocity: ( v = a t )\n- Distance traveled: ( s = \frac{1}{2} a t^2 )", "These equations are essential tools in both classroom physics and real-world aerospace engineering."]
