Hey guys! Ever wondered what it takes to navigate a spaceship around a planet? Let's dive into an exciting physics problem involving a spaceship maneuvering near Planet Zeta. We'll explore the concepts of position, velocity, acceleration, and time to figure out the spaceship's final position and velocity after a thruster burn. Get ready for a thrilling journey through the cosmos!
Initial Conditions: Setting the Stage
To begin this cosmic journey, let's define the initial conditions of our spaceship. Imagine our spaceship is cruising near Planet Zeta, a celestial body far, far away. Initially, the spaceship's position (r) relative to the planet is given as (600î - 400ĵ + 200k) x 10^3 km. This means the spaceship is 600,000 km in the î direction, -400,000 km in the ĵ direction, and 200,000 km in the k direction relative to Planet Zeta. Think of î, ĵ, and k as the x, y, and z axes in our cosmic coordinate system. This initial position is crucial for determining where the spaceship starts its journey. This initial placement is akin to setting the stage for an epic performance, providing the context for all subsequent movements and actions. Without this foundational understanding of where we begin, charting our course through the vastness of space would be akin to sailing without a map. Understanding the initial position allows us to contextualize the spaceship's movements, ensuring we can accurately predict its trajectory and interactions within the complex gravitational landscape surrounding Planet Zeta.
The spaceship's initial velocity (v) is 9500î m/s, meaning it's traveling at a brisk 9500 meters per second in the î direction. Velocity, as we know, is not just speed; it includes direction too. In this case, the spaceship is moving primarily along our defined 'x' axis. The initial velocity is the spacecraft's momentum at the starting point, the catalyst for its voyage through space. This speed, combined with the direction, paints a vivid picture of the spaceship's immediate path. Like an arrow launched from a bow, the initial velocity sets the tone for the entire flight, influencing how the spaceship interacts with gravitational forces and other celestial phenomena. Understanding this initial velocity is fundamental for predicting the spacecraft's immediate movements and trajectory alterations, a key component in ensuring a successful mission around Planet Zeta.
The Thruster Burn: Acceleration in Action
Now, for the exciting part! The spaceship fires up its thruster engine, initiating an acceleration (a) of (40î - 20k) m/s². This acceleration has components in both the î and k directions. The 40î component means the spaceship is accelerating at 40 meters per second squared in the î direction, while the -20k component means it's accelerating at -20 meters per second squared in the k direction (opposite to the k direction). This acceleration is the pivotal force driving changes in the spaceship's motion. It’s the energetic push that alters both the speed and direction, setting the spacecraft on a new course within the cosmic expanse. The firing of the thrusters is akin to a sculptor chiseling away at a block of marble, refining the spaceship's path with each precise adjustment. Comprehending the acceleration's magnitude and direction allows us to foresee how the spaceship's velocity will evolve over time, a critical calculation for navigating the complexities of space travel around Planet Zeta.
The thruster burns for a duration of 35 minutes. Time is a crucial factor here because acceleration acts over time to change velocity. The longer the thruster burns, the greater the change in velocity. This duration of 35 minutes is the temporal canvas upon which the acceleration paints its masterpiece of motion. It's the extended period during which the thrusters exert their influence, gradually sculpting the spaceship's trajectory. Think of this time as the steady hand of a conductor leading an orchestra, guiding the spaceship through a symphony of movement. Understanding this duration is paramount, as it directly correlates with the magnitude of the velocity change. Without this temporal dimension, we couldn't accurately predict how the spaceship's speed and direction will evolve, making precise navigation near Planet Zeta an impossibility.
Calculating the Final Velocity: Speed and Direction
Our first task is to determine the final velocity of the spaceship. We know that acceleration is the rate of change of velocity over time. So, we can use the following equation:
v_f = v_i + a * t
Where:
- v_f is the final velocity
- v_i is the initial velocity
- a is the acceleration
- t is the time
But before we plug in the values, let's make sure our units are consistent. We have time in minutes, but we need it in seconds since our acceleration is in meters per second squared. So, 35 minutes * 60 seconds/minute = 2100 seconds. This unit conversion is a crucial step, ensuring we're speaking the same mathematical language across all parameters. It's akin to translating measurements from inches to centimeters, ensuring a cohesive understanding of the dimensions at hand. Without this meticulous attention to unit consistency, our calculations would be skewed, leading to inaccurate predictions of the spaceship's motion. This meticulous approach guarantees that our final velocity calculations align perfectly with the physical realities of the spaceship's journey near Planet Zeta.
Now we can plug in the values:
v_f = 9500î m/s + (40î - 20k) m/s² * 2100 s
v_f = 9500î m/s + (84000î - 42000k) m/s
v_f = (9500 + 84000)î - 42000k m/s
v_f = 93500î - 42000k m/s
So, the final velocity of the spaceship is 93500 m/s in the î direction and -42000 m/s in the k direction. This final velocity unveils the spaceship's ultimate speed and direction after the thruster burn, revealing the cumulative impact of acceleration over time. It's the grand finale of our velocity calculation, painting a vivid picture of the spacecraft's post-maneuver momentum. Understanding the final velocity is pivotal for subsequent trajectory planning, ensuring the spaceship remains on course for its intended destination near Planet Zeta. This crucial parameter acts as the cornerstone for all future navigation decisions, guaranteeing a smooth and efficient journey through the cosmos.
Calculating the Final Position: Where Did We End Up?
Next, we need to find the final position of the spaceship. We can use the following kinematic equation:
r_f = r_i + v_i * t + 0.5 * a * t²
Where:
- r_f is the final position
- r_i is the initial position
- v_i is the initial velocity
- a is the acceleration
- t is the time
Plugging in the values, remembering to convert our initial position to meters:
r_f = (600 * 10^6 î - 400 * 10^6 ĵ + 200 * 10^6 k) m + (9500î m/s) * 2100 s + 0.5 * (40î - 20k) m/s² * (2100 s)²
r_f = (600 * 10^6 î - 400 * 10^6 ĵ + 200 * 10^6 k) m + (19950000î) m + 0.5 * (40î - 20k) m/s² * 4410000 s²
r_f = (600 * 10^6 î - 400 * 10^6 ĵ + 200 * 10^6 k) m + (19950000î) m + (88200000î - 44100000k) m
r_f = (600000000î - 400000000ĵ + 200000000k) m + 19950000î m + (88200000î - 44100000k) m
r_f = (600000000 + 19950000 + 88200000)î - 400000000ĵ + (200000000 - 44100000)k m
r_f = 708150000î - 400000000ĵ + 155900000k m
Converting back to kilometers:
r_f = (708.15î - 400ĵ + 155.9k) x 10^3 km
Therefore, the final position of the spaceship is (708.15î - 400ĵ + 155.9k) x 10^3 km relative to Planet Zeta. This final position reveals the spaceship's ultimate location after the thruster burn, showcasing the culmination of its initial trajectory, the applied acceleration, and the passage of time. It's the destination point on our cosmic map, the result of meticulous calculations and the physical laws governing motion. Understanding the final position is critical for ensuring the spaceship's successful navigation within the complex gravitational field surrounding Planet Zeta. This endpoint serves as the cornerstone for future maneuvers and trajectory adjustments, ensuring the mission remains on course and achieves its celestial objectives.
Conclusion: A Successful Maneuver
So, after burning its thrusters for 35 minutes, the spaceship's final velocity is 93500î - 42000k m/s, and its final position is (708.15î - 400ĵ + 155.9k) x 10^3 km relative to Planet Zeta. We successfully navigated the spaceship around Planet Zeta! Isn't physics awesome? We've journeyed through the cosmos using the principles of motion, transforming abstract equations into a tangible understanding of a spaceship's trajectory. This exploration highlights the power of physics in deciphering the complexities of space travel, from the initial push of the thrusters to the final positioning near a distant planet. The successful maneuver is a testament to the precision and predictability offered by physics, paving the way for future cosmic endeavors and further exploration of the universe.