Car's Angular Velocity: 200m In 5s

Hey guys, ever wondered if your car could actually pull off a perfectly uniform 200-meter dash in just 5 seconds? And what about figuring out the angular velocity of those tires while it's happening? It sounds like something straight out of a physics problem, and guess what? It totally is! We're diving deep into the world of motion, speed, and rotation to see if this scenario is even possible and, if it is, what kind of spin those wheels are putting in. Get ready to wrap your heads around some cool physics concepts, because we're going to break it all down, nice and easy.

First off, let's talk about the uniform motion part. When we say uniform motion, we're talking about a constant speed. No speeding up, no slowing down. This means the car is cruising along at the exact same pace for the entire 200 meters. So, the big question is: can a car cover that distance in that short amount of time uniformly? To figure this out, we need to calculate the car's linear velocity. The formula is super simple: velocity = distance / time. In our case, the distance is 200 meters and the time is 5 seconds. So, plug those numbers in: velocity = 200 m / 5 s. That gives us a linear velocity of 40 meters per second (m/s). Now, 40 m/s is pretty darn fast! To put that in perspective, that's about 144 kilometers per hour (km/h) or roughly 90 miles per hour (mph). While many cars can reach these speeds, maintaining such a high speed uniformly for a whole 200 meters right from a standstill might be a stretch in real-world driving conditions due to factors like acceleration and braking. However, for the sake of this physics problem, we're going to assume this uniform motion is achievable. The core idea here is understanding that uniformity implies a constant rate, and we've just calculated the required rate.

Unpacking the Physics: Linear vs. Angular Velocity

Alright, so we've got our linear velocity sorted out – it's 40 m/s. But the question isn't just about how fast the car is moving in a straight line; it's also asking about the angular velocity of its tires. This is where things get really interesting, guys! Think about it: as the car moves forward, its wheels are spinning. Linear velocity is how fast the car covers ground, while angular velocity is how fast the wheel is rotating. They are directly related, and that's where the tire's radius comes into play. Imagine a point on the edge of the tire. As the tire makes one full rotation, that point travels a distance equal to the circumference of the tire. The circumference is calculated using the formula $C = 2 \pi r$, where 'r' is the radius. In our problem, the radius of the tires is given as 0.4 meters. So, the circumference of each tire is $C = 2 \pi (0.4 \text{ m}) = 0.8 \pi \text{ meters}$. This means for every complete revolution, the tire (and thus the car) moves forward by $0.8 \pi$ meters. Now, how do we connect this to our linear velocity of 40 m/s? The relationship between linear velocity ($v$) and angular velocity ($\\omega$) is given by the formula $v = r \\omega$. Here, $v$ is the linear velocity, $r$ is the radius, and $\\omega$ is the angular velocity. We know $v$ and $r$, so we can rearrange the formula to solve for $\\omega$: $\\omega = v / r$. This is the key equation that links the car's forward motion to the spinning of its tires. It tells us that for a given linear speed, a smaller radius means faster spinning, and a larger radius means slower spinning. It's all about how much distance is covered per rotation.

Calculating Angular Velocity: The Math Behind the Spin

Now for the fun part – crunching the numbers to find that angular velocity! We've already established our key players: the linear velocity ($v$) of the car is 40 m/s, and the radius ($r$) of the tires is 0.4 meters. Our goal is to find the angular velocity ($\\omega$), which tells us how fast the tires are spinning in radians per second. Remember the formula we just talked about? It's $v = r \\omega$. To isolate $\\omega$, we simply divide both sides by $r$: $\\omega = v / r$. Let's plug in our values: $\\omega = 40 \text{ m/s} / 0.4 \text{ m}$. Performing the division, we get $\\omega = 100$ radians per second. So, the angular velocity of the tires is 100 radians per second. That's a pretty rapid spin! To give you a better sense of this, one full rotation is $2 \pi$ radians (approximately 6.28 radians). So, 100 radians per second means the tire is completing roughly $100 / (2 \pi) \approx 15.9$ full rotations every single second. Imagine that! It's a testament to how quickly things have to move when a car is traveling at 40 m/s. This calculation confirms that the physics allows for a direct relationship between the car's forward motion and the rotational speed of its wheels, provided the conditions like uniform motion and tire radius are met. It’s a neat way to see how different aspects of motion are interconnected in the world of physics.

Is it Possible? A Real-World Check

Let's get real for a second, guys. While the math works out perfectly, is it actually possible for a standard car to achieve a uniform 200 meters in 5 seconds? As we calculated, this requires a constant speed of 40 m/s, which is about 90 mph. Most production cars can reach 90 mph, but hitting that speed instantaneously and maintaining it uniformly for 200 meters is another story. The acceleration phase is crucial. Cars need time and distance to get up to speed. Think about it: when you floor it, there's a noticeable lag before you reach top speed. This lag is due to the car's engine power, transmission, weight, and aerodynamic drag. To cover 200 meters in 5 seconds, a car would need to accelerate incredibly quickly. For example, if a car accelerated uniformly from 0 m/s to 40 m/s in 5 seconds, its average speed would be $(0 + 40) / 2 = 20$ m/s, and it would cover a distance of $20 \text{ m/s} * 5 \text{ s} = 100$ meters. To cover 200 meters in 5 seconds, it implies the car is already at 40 m/s at the start or accelerates much faster than a typical car. The concept of uniform motion implies no acceleration. So, in a strict physics sense, for the motion to be uniform, the car would need to be traveling at 40 m/s from the very beginning of the 200 meters. This is highly unrealistic for a car starting from a standstill. However, if we interpret the question as