Distance Traveled Car Accelerating From 10 M/s To 30 M/s In 4 Seconds

Have you ever wondered how we calculate the distance a car travels while accelerating? It's a fascinating blend of physics and math! Let's dive into a classic problem: A car starts with a velocity of 10 m/s and reaches 30 m/s in 4 seconds. What distance did it cover, and what type of motion is it describing? This question is a perfect example of uniformly accelerated motion, a fundamental concept in physics. We'll break down the problem step-by-step, making it super easy to understand, even if you're not a physics whiz.

Understanding Uniformly Accelerated Motion

First, let's get the basics down. Uniformly accelerated motion occurs when an object's velocity changes at a constant rate. In simpler terms, the object speeds up or slows down evenly. Think of a car smoothly accelerating on a highway or a ball rolling down a ramp. The key here is the constant rate of change in velocity, which we call acceleration.

To tackle problems involving uniformly accelerated motion, we use a set of equations that relate displacement, initial velocity, final velocity, acceleration, and time. These equations are our trusty tools for solving a variety of motion-related puzzles. Before we jump into the calculations, let’s identify the information we already have from the problem statement:

• Initial velocity (v₀): 10 m/s (This is the car's starting speed)
• Final velocity (v): 30 m/s (The car's speed after accelerating)
• Time (t): 4 seconds (The duration of the acceleration)

Our mission is to find the distance (d) traveled by the car during these 4 seconds and to describe the type of motion. Now that we have our givens, let's explore the equations that will help us crack this problem!

The Magic Equations of Motion

There are three primary equations of motion that are crucial for solving problems involving uniform acceleration:

1. v = v₀ + at (This equation relates final velocity, initial velocity, acceleration, and time)
2. d = v₀t + (1/2)at² (This equation connects displacement, initial velocity, time, and acceleration)
3. v² = v₀² + 2ad (This equation links final velocity, initial velocity, acceleration, and displacement)

Where:

• v = final velocity
• v₀ = initial velocity
• a = acceleration
• t = time
• d = displacement (distance)

These equations might look a bit intimidating at first, but trust me, they're not as scary as they seem! The key is to choose the right equation for the problem at hand. To decide which equation to use, we look at the information we have (our givens) and what we're trying to find. In this case, we know v₀, v, and t, and we want to find d. So, we need an equation that includes these variables.

Looking at our equations, the second equation, d = v₀t + (1/2)at², seems promising. However, it also includes acceleration (a), which we don't know yet. So, we'll need to find acceleration first. Let's see how we can do that.

Calculating Acceleration

To find the acceleration, we can use the first equation of motion: v = v₀ + at. This equation is perfect because it directly relates final velocity (v), initial velocity (v₀), acceleration (a), and time (t), all of which we either know or want to find. Let's plug in the values we know:

30 m/s = 10 m/s + a(4 s)

Now, we can solve for acceleration (a). First, subtract 10 m/s from both sides of the equation:

20 m/s = a(4 s)

Next, divide both sides by 4 seconds:

a = (20 m/s) / (4 s) = 5 m/s²

So, the acceleration of the car is 5 m/s². This means the car's velocity is increasing by 5 meters per second every second. Now that we have the acceleration, we're one step closer to finding the distance traveled. Let's use this value in our next calculation.

Finding the Distance Traveled

Now that we know the acceleration (a = 5 m/s²), we can use the second equation of motion to find the distance (d) traveled by the car: d = v₀t + (1/2)at². Let's plug in the values we have:

d = (10 m/s)(4 s) + (1/2)(5 m/s²)(4 s)²

First, let's simplify the equation by performing the multiplications:

d = 40 m + (1/2)(5 m/s²)(16 s²)

Next, let's multiply (1/2) by 5 m/s² and then by 16 s²:

d = 40 m + (2.5 m/s²)(16 s²)

d = 40 m + 40 m

Finally, add the two terms together:

d = 80 m

Therefore, the car traveled a distance of 80 meters during the 4 seconds of acceleration. We've successfully calculated the distance using the equations of motion. But we're not done yet! We still need to describe the type of motion the car is undergoing.

Describing the Type of Motion

As we discussed earlier, this problem describes uniformly accelerated motion. This is because the car's velocity is changing at a constant rate, which is indicated by the constant acceleration we calculated (5 m/s²). In other words, the car is speeding up smoothly and consistently.

If the acceleration were zero, the car would be moving at a constant velocity, which is called uniform motion. If the acceleration were changing, the motion would be non-uniformly accelerated, which is a more complex scenario. But in this case, we have a clear example of uniformly accelerated motion, making our analysis straightforward and satisfying.

Putting It All Together

Let's recap what we've done. We started with a problem: a car accelerating from 10 m/s to 30 m/s in 4 seconds. We wanted to find the distance traveled and describe the type of motion. We identified the given information (initial velocity, final velocity, and time) and used the equations of motion to solve for the unknowns.

First, we calculated the acceleration using the equation v = v₀ + at, which gave us a = 5 m/s². Then, we used the equation d = v₀t + (1/2)at² to find the distance traveled, which turned out to be 80 meters. Finally, we identified the motion as uniformly accelerated motion because the car's velocity was changing at a constant rate.

This problem demonstrates the power of physics equations in describing and predicting real-world motion. By understanding the concepts of uniformly accelerated motion and the relationships between displacement, velocity, acceleration, and time, we can solve a wide range of problems. So, the next time you see a car speeding up, you'll have a better understanding of the physics at play!

Real-World Applications and Further Exploration

The concepts we've discussed here aren't just theoretical; they have countless applications in the real world. Understanding uniformly accelerated motion is crucial in fields like engineering, where designing vehicles, analyzing projectile motion, and optimizing sports equipment all rely on these principles.

For instance, when designing a car's braking system, engineers need to calculate the stopping distance based on the car's initial speed and deceleration rate. This involves the same equations we used to solve our problem. Similarly, when launching a satellite into orbit, scientists and engineers need to carefully calculate the trajectory and acceleration to ensure the satellite reaches its intended destination.

If you're interested in exploring this topic further, there are many avenues to pursue. You could delve deeper into the equations of motion, investigate non-uniformly accelerated motion, or explore the effects of air resistance and friction on moving objects. There are also numerous online resources, textbooks, and physics courses that can help you expand your knowledge.

Practice Problems to Sharpen Your Skills

To solidify your understanding of uniformly accelerated motion, try solving some practice problems. Here's one to get you started:

• A train accelerates from rest to 20 m/s in 10 seconds. What is the acceleration of the train, and how far does it travel during this time?

Work through this problem step-by-step, using the same approach we used for the car problem. Identify the givens, choose the appropriate equations, and solve for the unknowns. The more you practice, the more comfortable you'll become with these concepts.

Physics is a fascinating subject that helps us understand the world around us. By mastering the basics of motion, you'll gain a deeper appreciation for the principles that govern our universe. So keep exploring, keep questioning, and keep learning!

So, in summary, the car traveled 80 meters while undergoing uniformly accelerated motion. We successfully solved the problem by applying the fundamental equations of motion and understanding the concepts of initial velocity, final velocity, acceleration, time, and displacement. Physics can be fun, right? Keep exploring and stay curious, guys!