Speed Calculation: Determining If A Car Exceeded Speed Limit

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Hey guys! Today, we're diving into a cool physics problem that involves calculating speed and figuring out if a car was speeding. This is super practical because it's exactly what traffic police do with radar guns! Let's break down the problem step by step and make sure we understand everything.

The Problem: Speed Limit and Car's Speed

Okay, so here’s the situation. We know the legal speed limit outside urban areas is v₁ = 90 km/h. Our mission is to:

  • Figure out the car's speed if the radar detects a displacement of d = 60 cm in a time interval of t = 1/50 s.
  • Determine if the car exceeded the speed limit.

Sounds like a fun challenge, right? Let’s get started!

Step 1: Understanding the Basics of Speed

Before we jump into calculations, let’s quickly recap what speed actually means. Speed is essentially how fast an object is moving. In physics terms, it’s the rate at which an object covers distance. The formula we use to calculate speed is:

Speed = Distance / Time

This formula is your best friend in these kinds of problems. It tells us that if we know the distance an object traveled and the time it took, we can easily find its speed. Simple, right?

Why Units Matter: Kilometers per Hour vs. Meters per Second

Now, a crucial thing to keep in mind is units. In this problem, we’re given the speed limit in kilometers per hour (km/h), but the displacement is in centimeters (cm) and time is in seconds (s). This mismatch can cause big problems if we don't handle it correctly. We need to make sure all our units are consistent before we start plugging numbers into the formula. The most common way to do this is to convert everything to the SI units, which are meters (m) for distance and seconds (s) for time. This will give us the speed in meters per second (m/s).

Converting Units: A Quick How-To

So, how do we convert km/h to m/s and cm to meters? Let’s break it down:

  • Kilometers per hour (km/h) to meters per second (m/s): To convert from km/h to m/s, we multiply by 1000/3600 (since there are 1000 meters in a kilometer and 3600 seconds in an hour). This simplifies to dividing by 3.6.

    1 km/h = (1000 m) / (3600 s) = 1/3.6 m/s

  • Centimeters (cm) to meters (m): To convert centimeters to meters, we divide by 100 (since there are 100 centimeters in a meter).

    1 cm = 1/100 m = 0.01 m

Understanding these conversions is super important for getting the right answer. Always double-check your units before you start calculating!

Step 2: Converting Given Values to SI Units

Alright, let’s get those values converted! We have:

  • Displacement, d = 60 cm
  • Time, t = 1/50 s
  • Speed limit, v₁ = 90 km/h

First, let’s convert the displacement from centimeters to meters:

d = 60 cm = 60 / 100 m = 0.6 m

Easy peasy! Now, let’s convert the speed limit from kilometers per hour to meters per second:

v₁ = 90 km/h = 90 / 3.6 m/s = 25 m/s

So, the legal speed limit is 25 meters per second. We've got all our values in the right units now, which means we’re ready to roll!

Step 3: Calculating the Car's Speed

Now for the fun part – calculating the car's speed! We’ll use our trusty speed formula:

Speed = Distance / Time

We know the distance (d = 0.6 m) and the time (t = 1/50 s), so let’s plug those values in:

Speed = 0.6 m / (1/50 s)

To divide by a fraction, we multiply by its reciprocal. So, we get:

Speed = 0.6 m * (50/1 s) = 0.6 * 50 m/s = 30 m/s

The car's speed is 30 meters per second. We're almost there!

Step 4: Determining if the Car Exceeded the Speed Limit

Okay, we’ve got the car's speed (30 m/s) and the speed limit (25 m/s). Now we just need to compare them. Is 30 m/s greater than 25 m/s? You bet it is!

Car's speed (30 m/s) > Speed limit (25 m/s)

This means the car exceeded the speed limit. Busted!

Step 5: Summarizing the Solution and Emphasizing Key Steps

Let's quickly recap what we've done and why each step is important. First, we understood the problem, which is crucial. Then, we made sure to convert all the values to consistent units. This is a step that’s easy to overlook but can totally mess up your answer if you skip it. We then applied the speed formula, Speed = Distance / Time, which is a fundamental concept in physics. Finally, we compared the calculated speed with the speed limit to determine if the car was speeding.

Here’s a quick summary of our calculations:

  1. Converted 60 cm to 0.6 m.
  2. Converted 90 km/h to 25 m/s.
  3. Calculated the car's speed: Speed = 0.6 m / (1/50 s) = 30 m/s.
  4. Compared the car's speed (30 m/s) to the speed limit (25 m/s) and found that the car exceeded the limit.

Practical Implications and Real-World Applications

This problem isn't just a theoretical exercise; it has real-world applications. Understanding how to calculate speed and convert units is essential in many fields, from physics and engineering to everyday situations like driving. Radar guns used by traffic police rely on these same principles to measure the speed of vehicles. By calculating speed accurately, we can ensure road safety and prevent accidents.

Conclusion: Speeding is a No-Go!

So, there you have it! We successfully calculated the car's speed and determined that it was indeed speeding. Remember, guys, speed limits are there for a reason – to keep everyone safe. By understanding the physics behind speed calculations, we can make better, safer decisions on the road. Keep practicing these kinds of problems, and you’ll become a physics whiz in no time!

If you found this helpful, give it a thumbs up and share it with your friends. And if you have any questions or want to dive deeper into physics, drop a comment below. Until next time, stay safe and keep learning!