Calculate Acceleration Uniformly - Step-by-Step Solution

Hey guys! Today, we're diving into a super cool physics problem that involves calculating the acceleration of a car. This is a classic example that you'll often encounter in introductory physics courses, and it's a fantastic way to understand the fundamental concepts of motion. So, let's break it down step by step and make sure we grasp every detail. We're going to explore how to find the acceleration of a car that starts from a standstill and speeds up to 72 meters per second in just 8 seconds. Sounds like a fun ride, right? Buckle up, and let's get started!

The problem we're tackling today involves a car that starts from a complete stop – that's what we mean by "rest." This car then accelerates uniformly, which is a fancy way of saying it speeds up at a consistent rate. It takes the car 8 seconds to reach a velocity of 72 meters per second. Our mission, should we choose to accept it, is to calculate the value of this acceleration. To make sure we're all on the same page, let's quickly define some key terms. Acceleration, in simple terms, is how quickly the velocity of an object changes. Velocity, on the other hand, is the speed of an object in a given direction. When we say the car accelerates uniformly, it means the velocity is changing at a steady rate. This makes our calculations much easier because we can use straightforward formulas to find the acceleration. So, the core of the problem boils down to understanding how these concepts fit together and applying the right formula to solve for acceleration.

Alright, before we jump into calculations, let's make sure we've got all our ducks in a row. The first crucial step in solving any physics problem is to identify the given values. This helps us organize our thoughts and see what tools we have at our disposal. In this scenario, we know a few key pieces of information. First, the car starts from rest. This means its initial velocity (let's call it v₀) is 0 meters per second. Next, we know the car takes 8 seconds to reach a certain speed. So, our time interval (t) is 8 seconds. Lastly, the car ends up traveling at a velocity of 72 meters per second. This is our final velocity (v). By clearly identifying these values – v₀ = 0 m/s, t = 8 s, and v = 72 m/s – we've laid a solid foundation for solving the problem. It's like gathering the ingredients before you start cooking; you need to know what you have to work with! Now that we have these values, we can move on to the next step: choosing the right formula.

Okay, so now that we've got our given values sorted out, the next big question is: what formula do we use to find the acceleration? Don't worry, it's not as scary as it sounds! When dealing with uniform acceleration, there's a go-to formula that's super handy: a = (v - v₀) / t. Let's break this down. a stands for acceleration (which is what we're trying to find), v is the final velocity, v₀ is the initial velocity, and t is the time interval. This formula basically says that acceleration is the change in velocity divided by the time it took for that change to happen. It's a simple yet powerful tool for solving problems like this one. Why is this the right formula? Well, it directly relates the quantities we know (initial velocity, final velocity, and time) to the quantity we want to find (acceleration). Other formulas might involve distance or other variables we don't have in this problem. So, this one fits perfectly. Once you've identified the right formula, the rest is just plugging in the numbers and doing the math. Let's move on to that next!

Alright, guys, the moment we've been waiting for! Now it's time to plug those numbers we identified earlier into our chosen formula. Remember, we're using a = (v - v₀) / t. We know that the final velocity (v) is 72 meters per second, the initial velocity (v₀) is 0 meters per second (since the car started from rest), and the time (t) is 8 seconds. So, let's slot those values into the equation. We get: a = (72 m/s - 0 m/s) / 8 s. See how we're just replacing the symbols with their corresponding values? This is a crucial step in solving any physics problem. It turns an abstract formula into a concrete calculation. By plugging in the values carefully, we avoid making silly mistakes and set ourselves up for a correct answer. It's like putting the right ingredients into a recipe – you need the right amounts to get the delicious result you're aiming for. So, let's take a deep breath and double-check that we've got everything in the right place. Once we're confident, we can move on to the actual calculation.

Okay, let's crunch some numbers! We've plugged our values into the formula a = (72 m/s - 0 m/s) / 8 s. The next step is to simplify this equation and get our answer. First, let's take care of the numerator: 72 m/s minus 0 m/s is simply 72 m/s. So, our equation now looks like this: a = 72 m/s / 8 s. Now, we just need to divide 72 by 8. If you're quick with your times tables, you'll know that 72 divided by 8 is 9. So, a = 9 m/s². And there you have it! We've calculated the acceleration. It's really that straightforward once you've got the formula and the values plugged in. The key here is to take it one step at a time, simplify as you go, and don't rush the process. A little bit of careful arithmetic can make all the difference. Now that we have our numerical answer, there's one more important thing to do: think about the units and make sure they make sense.

Alright, we've got our answer: a = 9. But we're not quite done yet! It's super important to include the units to give our answer meaning. In this case, we're calculating acceleration, and the standard unit for acceleration is meters per second squared, or m/s². So, our final answer is a = 9 m/s². But what does this actually mean? The m/s² unit tells us how much the velocity changes every second. An acceleration of 9 m/s² means that the car's velocity increases by 9 meters per second every second. Think about it this way: at the start, the car is at rest (0 m/s). After one second, it's going 9 m/s. After two seconds, it's going 18 m/s, and so on. This gives us a clear picture of how quickly the car is speeding up. Understanding the units and what they represent is crucial for interpreting our results correctly. It's not just about getting the right number; it's about understanding what that number means in the real world. So, always remember to include your units and take a moment to think about what they tell you!

Okay, guys, let's bring it all home! After carefully working through the problem, plugging in our values, and crunching the numbers, we've arrived at our final answer. The acceleration of the car is 9 meters per second squared (9 m/s²). Let's recap what that means: the car's velocity increases by 9 meters per second every second it accelerates. This is a constant, uniform acceleration, meaning the car speeds up smoothly and consistently. We've successfully calculated the acceleration using the formula a = (v - v₀) / t, where v is the final velocity, v₀ is the initial velocity, and t is the time interval. By identifying the given values, choosing the correct formula, plugging in the numbers, and paying attention to units, we've conquered this physics problem. This kind of step-by-step approach is super helpful for tackling any physics question. So, remember these steps, and you'll be well on your way to mastering motion problems. Great job, everyone! We nailed it!

Understanding the Problem

The question asks us to calculate the acceleration of a car that starts from rest and reaches a velocity of 72 m/s in 8 seconds. The key here is the phrase "uniform acceleration," which means the car's velocity increases at a constant rate. To solve this, we'll use a fundamental physics formula that relates acceleration, initial velocity, final velocity, and time.

Key Concepts

• Acceleration: The rate at which an object's velocity changes over time. It is measured in meters per second squared (m/s²).
• Initial Velocity (v₀): The velocity of the object at the beginning of the time interval. In this case, the car starts from rest, so v₀ = 0 m/s.
• Final Velocity (v): The velocity of the object at the end of the time interval. Here, the final velocity is 72 m/s.
• Time (t): The duration of the time interval, which is 8 seconds in this problem.

Steps to Calculate Acceleration

To find the acceleration, we'll follow these steps:

1. Identify the given values: List all the known quantities from the problem statement.
2. Choose the appropriate formula: Select the formula that relates acceleration, initial velocity, final velocity, and time.
3. Plug in the values: Substitute the known quantities into the formula.
4. Perform the calculation: Solve the equation for acceleration.
5. State the answer with units: Express the final answer with the correct units (m/s²).

Step 1: Identify the Given Values

From the problem statement, we have:

• Initial velocity (v₀) = 0 m/s (since the car starts from rest)
• Final velocity (v) = 72 m/s
• Time (t) = 8 s

Step 2: Choose the Appropriate Formula

The formula that relates acceleration (a), initial velocity (v₀), final velocity (v), and time (t) is:

a = (v - v₀) / t

This formula is derived from the definition of acceleration as the change in velocity divided by the change in time. It is a fundamental equation in kinematics, the study of motion.

Step 3: Plug In the Values

Now, we'll substitute the known values into the formula:

a = (72 m/s - 0 m/s) / 8 s

Step 4: Perform the Calculation

Next, we'll simplify the equation and perform the division:

a = (72 m/s) / 8 s

a = 9 m/s²

Step 5: State the Answer with Units

Finally, we state the answer with the correct units:

The acceleration of the car is 9 m/s².

Explanation of the Result

The acceleration of 9 m/s² means that the car's velocity increases by 9 meters per second every second. In other words:

• At t = 0 s, the car's velocity is 0 m/s.
• At t = 1 s, the car's velocity is 9 m/s.
• At t = 2 s, the car's velocity is 18 m/s.
• At t = 3 s, the car's velocity is 27 m/s.
• And so on, until it reaches 72 m/s at t = 8 s.

This constant increase in velocity is what we mean by uniform acceleration.

Why This Formula Works

The formula a = (v - v₀) / t is a direct application of the definition of acceleration. Acceleration is the rate of change of velocity, which is calculated as the change in velocity (v - v₀) divided by the time interval (t). This formula holds true when the acceleration is constant, which is the case in this problem.

Common Mistakes to Avoid

When solving problems involving acceleration, there are a few common mistakes to watch out for:

• Forgetting the Units: Always include the units in your calculations and final answer. Acceleration is measured in m/s².
• Incorrectly Identifying Initial and Final Velocities: Make sure you correctly identify which velocity is the initial velocity (v₀) and which is the final velocity (v).
• Using the Wrong Formula: Use the correct formula for the given situation. In this case, the formula a = (v - v₀) / t is appropriate because we know the initial velocity, final velocity, and time.
• Arithmetic Errors: Double-check your calculations to avoid making arithmetic errors.

Conclusion

We have successfully calculated the acceleration of the car using the formula a = (v - v₀) / t. By identifying the given values, plugging them into the formula, and performing the calculation, we found that the acceleration is 9 m/s². Remember to always include units and interpret your results in the context of the problem. Understanding these fundamental physics concepts and practicing problem-solving techniques will help you master mechanics and other areas of physics. Keep practicing, and you'll become a pro at solving these types of problems!

To solidify your understanding of acceleration, try solving these practice problems:

1. A bicycle accelerates from rest to a velocity of 15 m/s in 5 seconds. Calculate the acceleration.
2. A train traveling at 30 m/s decelerates to a stop in 10 seconds. Calculate the deceleration (negative acceleration).
3. A car accelerates from 10 m/s to 25 m/s in 6 seconds. Calculate the acceleration.

Solving these problems will give you more practice with the formula and help you understand the concept of acceleration better.

Calculate the acceleration of a car that starts from rest, reaches a speed of 72 m/s in 8 seconds, and accelerates uniformly.

Calculate Acceleration Uniformly - Step-by-Step Solution