Hey physics enthusiasts! Today, we're diving into a classic kinematics problem that involves calculating acceleration given some initial conditions. We'll use a well-known equation of motion to solve for the unknown. So, buckle up and let's get started!
We're given the equation of motion:
Where:
- s represents the displacement (1.00 km)
- u is the initial velocity (2 m/s)
- t denotes the time elapsed (1 hour)
- a is the acceleration, which we need to find.
Our task is to determine the value of a using the provided information. This is a fundamental problem in physics, often encountered when studying motion with constant acceleration. Mastering this type of problem is crucial for understanding more complex physics concepts later on. So, let's break it down step by step to ensure we grasp every detail. We'll start by converting all units to a consistent system, and then we'll rearrange the equation to isolate the acceleration variable. Finally, we'll plug in the values and calculate the result. Remember, physics is all about understanding the relationships between different quantities, and this problem is a perfect example of that. It's not just about the numbers; it's about the principles behind them. So, let's not rush through it. Instead, let's take our time and make sure we fully understand the process. This way, we'll be well-equipped to tackle similar problems in the future.
1. Unit Conversion
Before we can plug the values into the equation, we need to ensure that all the units are consistent. Let's convert kilometers to meters and hours to seconds.
- s = 1.00 km = 1.00 * 1000 m = 1000 m
- t = 1 hr = 1 * 60 * 60 s = 3600 s
Why is unit conversion so important? You might ask. Well, imagine trying to add apples and oranges – it just doesn't work! Similarly, in physics, if our units are mismatched, we'll get nonsensical results. By converting everything to a standard system (in this case, meters and seconds), we ensure that our calculations are accurate and meaningful. Think of it as speaking the same language as the equation; otherwise, it won't understand what we're trying to say. This step is often overlooked, especially by beginners, but it's a fundamental aspect of problem-solving in physics. It's like laying the foundation for a building; if the foundation isn't solid, the whole structure will be unstable. So, always double-check your units before you start plugging numbers into equations. It can save you a lot of headaches down the line. And remember, practice makes perfect! The more you work with unit conversions, the more natural they'll become. You'll start to see patterns and shortcuts, and soon you'll be converting units like a pro. So, keep practicing, keep asking questions, and never underestimate the power of consistent units!
2. Rearranging the Equation
Now, let's rearrange the equation to solve for a:
Multiply both sides by 2:
Rearrange to isolate the term with a:
Divide both sides by t²:
Rearranging equations is a crucial skill in physics and mathematics. It's like having a set of building blocks and figuring out how to put them together to create the structure you want. In this case, we started with the equation for displacement and manipulated it to isolate the variable we were interested in – acceleration. The process involves using algebraic operations like addition, subtraction, multiplication, and division to move terms around and get the desired variable by itself on one side of the equation. It might seem a bit daunting at first, but with practice, you'll develop a knack for it. Think of it as a puzzle; each step is a move that brings you closer to the solution. And just like with puzzles, there can be multiple ways to get to the answer. The key is to understand the rules (the algebraic operations) and to think logically about how to apply them. So, don't be afraid to experiment, try different approaches, and see what works. And remember, it's okay to make mistakes! Mistakes are opportunities to learn and improve. So, embrace them, analyze them, and use them to refine your skills. The more you practice rearranging equations, the more confident and proficient you'll become. It's a skill that will serve you well in all areas of physics and beyond.
3. Plugging in the Values
Let's substitute the known values into the rearranged equation:
Plugging in the values is where the theory meets the reality of the problem. We've done the groundwork: we've understood the equation, converted the units, and rearranged the formula to isolate the unknown variable. Now comes the exciting part – putting in the actual numbers and seeing what we get. It's like cooking a recipe; you've gathered all the ingredients and followed the instructions, and now you're ready to combine them and see the final dish. But just like in cooking, attention to detail is crucial. We need to make sure we're plugging in the right values in the right places. A small mistake here can lead to a big error in the final answer. So, it's always a good idea to double-check your work and make sure everything lines up. It's also important to pay attention to the units. We've already made sure they're consistent, but we need to carry them along with the numbers in our calculation. This will help us ensure that the final answer has the correct units as well. Plugging in the values is not just about crunching numbers; it's about connecting the abstract symbols in the equation to the concrete reality of the problem. It's about seeing how the different quantities interact and influence each other. And it's about getting one step closer to the solution. So, take your time, be meticulous, and enjoy the process of bringing the equation to life!
4. Calculation
Now, let's perform the calculation:
Performing the calculation is the moment of truth. It's where all our preparation and hard work come together to produce the final answer. It's like the climax of a story, where the suspense builds up and finally releases. But it's not just about getting the right number; it's also about understanding what that number means. In this case, we're calculating acceleration, which is the rate of change of velocity. A negative acceleration means that the object is slowing down, or decelerating. So, the number we get tells us not only how much the object is accelerating but also in what direction. The calculation itself might involve a few steps, like multiplication, division, addition, and subtraction. It's important to follow the order of operations (PEMDAS/BODMAS) to ensure we get the correct result. And it's always a good idea to use a calculator or other tool to help with the arithmetic, especially if the numbers are large or complex. But even with the help of a calculator, it's crucial to understand what we're doing and why. We should be able to explain each step in the calculation and how it contributes to the final answer. So, don't just blindly punch numbers into a calculator; think about what you're doing and make sure it makes sense. And when you get the final answer, take a moment to reflect on it. Does it seem reasonable? Does it fit with your intuition about the problem? If not, go back and check your work. The calculation is not just a mechanical process; it's an opportunity to deepen your understanding of the physics involved. So, embrace the challenge, be meticulous, and enjoy the satisfaction of arriving at the correct answer!
The acceleration a is approximately -0.000957 m/s². The negative sign indicates that the acceleration is in the opposite direction to the initial velocity, meaning the object is decelerating.
Interpreting the result is the final and perhaps the most crucial step in solving a physics problem. It's not enough to just get a number; we need to understand what that number means in the context of the problem. It's like reading a book and not just stopping at the last page, but also reflecting on the story and its message. In this case, we've calculated the acceleration, but what does -0.000957 m/s² actually tell us? The negative sign is a key piece of information. It indicates that the acceleration is in the opposite direction to the initial velocity. This means the object is slowing down, or decelerating. If the acceleration were positive, it would mean the object is speeding up. The magnitude of the acceleration, 0.000957 m/s², tells us how quickly the object's velocity is changing. In this case, it's a small number, which means the object is decelerating slowly. We can also compare this value to other accelerations we might encounter in everyday life. For example, the acceleration due to gravity is about 9.8 m/s², which is much larger than our calculated acceleration. This gives us a sense of the scale of the deceleration in our problem. Interpreting the result also involves checking if it makes sense in the real world. Does the answer seem reasonable given the initial conditions and the physical situation? If the answer seems wildly out of line, it might indicate a mistake in our calculations or a misunderstanding of the problem. So, always take the time to interpret your results and make sure they make sense. It's the final step in the problem-solving process, and it's what ties everything together.
We successfully calculated the acceleration a using the given equation and initial conditions. Remember, the key to solving physics problems is to break them down into smaller, manageable steps. Unit conversion, rearranging equations, plugging in values, and performing calculations are all essential skills. Keep practicing, and you'll become a physics pro in no time!
Wrapping up a physics problem is like putting the finishing touches on a work of art. We've gone through all the steps, from understanding the problem to interpreting the result, and now it's time to step back and admire the complete picture. It's an opportunity to consolidate our learning, reinforce the concepts, and appreciate the power of physics in describing the world around us. In this case, we've successfully calculated the acceleration of an object using the equation of motion. We've seen how unit conversion, equation manipulation, and careful calculation all contribute to the final answer. We've also learned the importance of interpreting the result and understanding its physical meaning. But the learning doesn't stop here. We can use this problem as a springboard to explore other related concepts. For example, we could investigate the relationship between acceleration and force, or we could consider how air resistance or friction might affect the motion of the object. We could also try solving similar problems with different initial conditions or different equations of motion. The possibilities are endless! Physics is a vast and fascinating field, and every problem we solve is a step further on the journey of discovery. So, keep asking questions, keep exploring, and keep pushing the boundaries of your understanding. And remember, the most important thing is not just to get the right answer, but to understand the process and the principles behind it. So, let's celebrate our success in solving this problem and use it as motivation to tackle even greater challenges in the future!