Deceleration Distance: A Physics Problem Solved

Hey everyone! Let's dive into a classic physics problem involving a vehicle changing its speed. We'll break down the steps to calculate the distance traveled when a car decelerates. This is a super common type of problem you'll see in physics, so understanding how to solve it is key!

Understanding the Problem: Initial Conditions

So, here's the scenario: Our mobile, let's call it a car for simplicity, starts off cruising at 80 km/h. That's our initial speed. But then, something happens – maybe the driver sees a speed limit change, a stop sign, or maybe just wants to slow down. The car then decides to slow its roll. And it decreases its speed to 30 km/h. This change in speed isn't instantaneous; it happens over a period of time. We're told this deceleration takes place over 50 seconds. The core question we're trying to answer is: How far does the car travel during those 50 seconds while it's slowing down? To solve this, we need to understand some key physics concepts and apply the right formulas. This isn't just about plugging in numbers; it's about understanding the motion of the car. The car’s motion in these 50 seconds is crucial to understanding. We’re not just interested in the starting and ending speeds, but also how the speed changes over time. This is where the concept of acceleration (or in this case, deceleration) comes into play. We need to figure out how quickly the car's speed is decreasing to accurately calculate the distance traveled. We'll need to use kinematics equations, which are the bread and butter of describing motion in physics. Before we can jump into those equations, though, we need to make sure our units are consistent. We're dealing with speeds in kilometers per hour and time in seconds, so a conversion is in order! We want to ensure that we're working with a consistent set of units throughout the calculation. This will prevent errors and make the final answer more meaningful. So, the initial setup is key. We've got our initial speed, our final speed, and the time it takes for the speed to change. The missing piece is the distance traveled during that deceleration. Let’s get this figured out!

Step 1: Converting Units – km/h to m/s

Alright, first things first: those kilometers per hour (km/h) need to become meters per second (m/s). Why? Because seconds are our unit of time, and meters are a standard unit of distance in physics. Using mixed units is like trying to mix oil and water – it just doesn't work! We need everything playing on the same field. So, how do we do this conversion? Here's the breakdown:

• Kilometers to meters: 1 kilometer (km) is equal to 1000 meters (m).
• Hours to seconds: 1 hour is equal to 60 minutes, and each minute has 60 seconds. So, 1 hour is 60 * 60 = 3600 seconds.

Now, let's apply this to our speeds:

• Initial speed (80 km/h): To convert this to m/s, we multiply by 1000 (to convert km to m) and divide by 3600 (to convert hours to seconds). So, 80 km/h * (1000 m/km) / (3600 s/h) = 22.22 m/s (approximately).
• Final speed (30 km/h): Similarly, 30 km/h * (1000 m/km) / (3600 s/h) = 8.33 m/s (approximately).

Important Note: Always round your conversions appropriately. In this case, two decimal places should be enough for our calculations. Why is this step so crucial? Well, if we try to use km/h and seconds in the same equation, we'll get a nonsensical result. The units won't cancel out properly, and our answer will be way off. Think of it like this: you can't add apples and oranges – you need to convert them to a common unit, like "fruit". The same goes for physics calculations. Consistent units are our best friends! Now that we've got our speeds in meters per second, we're one step closer to solving the problem. We've taken a potentially messy situation and made it much cleaner and easier to work with. Next up, we'll use these converted speeds to figure out the car's acceleration (or deceleration). So, stay tuned!

Step 2: Calculating Acceleration (Deceleration)

Okay, we've got our initial and final speeds in meters per second, and we know the time it took for the car to slow down. Now, we need to calculate the acceleration. But wait, the car is slowing down, so is it really acceleration? Yes and no. In physics, acceleration is the rate of change of velocity, which can be either speeding up OR slowing down. When something slows down, we often call it deceleration, but mathematically, it's just negative acceleration.

The Formula: The formula for acceleration is super important and one you'll use again and again in physics: Acceleration (a) = (Final Velocity (vf) - Initial Velocity (vi)) / Time (t). Let's break down what this means:

• Final Velocity (vf): This is the speed at the end of the time interval, in our case, 8.33 m/s.
• Initial Velocity (vi): This is the speed at the beginning of the time interval, which is 22.22 m/s.
• Time (t): This is the duration of the change in speed, which we know is 50 seconds.

Let's plug in the numbers: a = (8.33 m/s - 22.22 m/s) / 50 s. This gives us a = -13.89 m/s / 50 s. Finally, we get a = -0.2778 m/s². That's our acceleration! Notice the negative sign? That's crucial! It tells us the car is decelerating, meaning its velocity is decreasing over time. The units of acceleration are meters per second squared (m/s²), which might seem a little weird, but it makes sense. It's the rate of change of velocity (m/s) per second. Think of it as how many meters per second the speed changes every second. Why is calculating acceleration so important? Because it's a key piece of the puzzle in figuring out the distance traveled. We know how the car's speed is changing, and that, along with the time, will allow us to use another kinematic equation to find the distance. Also, it gives us more in-depth understanding of the car's motion. So, we've successfully calculated the deceleration of the car. We're on a roll! Now, let's use this information to finally calculate the distance the car travels while slowing down. This is the home stretch!

Step 3: Calculating the Distance Traveled

Alright, we're at the final step! We know the initial velocity, the final velocity, the time, and the acceleration (deceleration). Now, we need to find the distance the car traveled during those 50 seconds. For this, we'll use one of the fundamental kinematic equations that relates distance, initial velocity, final velocity, acceleration, and time. There are a few equations we could use, but the most convenient one for this problem is:

The Formula: Distance (d) = vi * t + 0.5 * a * t². Let's break this down again:

• Distance (d): This is what we're trying to find!
• vi: Initial velocity, which we know is 22.22 m/s.
• t: Time, which is 50 seconds.
• a: Acceleration (deceleration), which we calculated as -0.2778 m/s².

Time to plug in the values: d = (22.22 m/s) * (50 s) + 0.5 * (-0.2778 m/s²) * (50 s)². Let's do the math:

• (22.22 m/s) * (50 s) = 1111 meters
• 0. 5 * (-0.2778 m/s²) * (50 s)² = 0.5 * (-0.2778 m/s²) * (2500 s²) = -347.25 meters

Now, add those together: d = 1111 meters + (-347.25 meters) = 763.75 meters. So, there you have it! The car travels approximately 763.75 meters while slowing down from 80 km/h to 30 km/h over 50 seconds. Why does this equation work? It essentially combines the effect of the initial velocity carrying the car forward with the effect of the deceleration slowing it down. The first part (vi * t) is the distance the car would have traveled if it maintained its initial speed. The second part (0.5 * a * t²) accounts for the change in speed due to acceleration. This is a really powerful equation that shows how all these different aspects of motion are connected. We've successfully used our knowledge of physics to solve a real-world-ish problem. And that's what physics is all about! You've taken a word problem and broken it down into manageable steps, applying the correct concepts and equations along the way.

Final Answer and Recap

So, the final answer is that the car travels approximately 763.75 meters while decelerating. Woohoo! Let's recap the steps we took to get there:

1. Understood the problem: We carefully identified the given information (initial speed, final speed, time) and what we needed to find (distance).
2. Converted units: We changed km/h to m/s to ensure consistent units in our calculations. This is crucial for accuracy.
3. Calculated acceleration: We used the formula a = (vf - vi) / t to find the deceleration of the car.
4. Calculated distance: We used the kinematic equation d = vi * t + 0.5 * a * t² to determine the distance traveled during deceleration.

This type of problem is a classic example of how physics can be used to describe and predict motion. By understanding the concepts of velocity, acceleration, and distance, and how they relate to each other, we can solve a wide range of problems. Remember, the key is to break down the problem into smaller, manageable steps, identify the relevant information, and choose the appropriate formulas. And most importantly, don't be afraid to ask questions and practice! Physics can be challenging, but it's also incredibly rewarding. Understanding the world around us through the lens of physics is a powerful and fascinating thing. So, keep exploring, keep learning, and keep solving problems! You've got this!