Plane Takeoff Speed: Calculate Final Velocity

Oct 17, 2025 by ADMIN 46 views

Hey guys! Let's break down this physics problem about a plane taking off. We're going to figure out how fast the plane is going when it finally gets airborne. This is a classic kinematics problem, and understanding how to solve it can help you with all sorts of physics challenges. So, let's dive in!

Understanding the Problem

Okay, so here's the deal: a plane needs 20 seconds and 400 meters of runway to get off the ground. It starts from a standstill, meaning its initial speed is zero. The big question is: What's the plane's speed at the moment it takes off? This is what we need to calculate. To really nail this, we need to tap into some basic physics principles, specifically the equations of motion that describe how things move with constant acceleration. These equations are our toolkit for solving this kind of problem.

The key here is that the plane is accelerating at a constant rate. Think about it: it starts slow and gets faster and faster until it reaches takeoff speed. This constant acceleration allows us to use specific equations that relate distance, time, initial velocity, final velocity, and acceleration. By identifying what we know (the givens) and what we want to find, we can choose the right equation and plug in the numbers. We'll walk through this step-by-step, so don't worry if it seems a little confusing right now.

So, before we jump into the math, let's recap the important info. We know the time it takes for takeoff (20 seconds), the distance the plane travels (400 meters), and the initial velocity (0 m/s). We want to find the final velocity. Now that we've got a clear picture of the problem, we can start thinking about which equation will help us get to the answer. We will look at the formulas next to calculate the final velocity, guys!

Choosing the Right Equation

Alright, so which equation do we use? There are a few kinematics equations, but we need the one that connects distance, time, initial velocity, final velocity, and acceleration. The equation that fits the bill perfectly is:

d = v₀t + (1/2)at²

Where:

d = distance (400 meters)
v₀ = initial velocity (0 m/s)
t = time (20 seconds)
a = acceleration (what we need to find first)

But wait! We don't know the acceleration yet. That's okay! We can use this equation to find the acceleration first, and then we'll use another equation to find the final velocity. This is a common strategy in physics problems: sometimes you need to solve for one thing before you can solve for what you really want.

So, let's rearrange the equation to solve for acceleration (a). Since v₀ is 0, the term v₀t becomes 0, which simplifies things a lot. Our equation now looks like this:

d = (1/2)at²

Now, we can multiply both sides by 2 and divide by t² to isolate 'a':

a = 2d / t²

Now we have a formula to calculate the acceleration. We have all the values on hand, guys. We know the distance (d) is 400 meters, and the time (t) is 20 seconds. Once we calculate the acceleration, we can use another kinematic equation to find the final velocity. This step-by-step approach is super helpful in breaking down complex problems into smaller, manageable chunks. So, let's plug in those numbers and get that acceleration figured out!

Calculating Acceleration

Okay, let's plug in the values we know into the equation we just derived:

a = 2d / t²

We know that:

d = 400 meters
t = 20 seconds

So, substituting these values in, we get:

a = (2 * 400) / (20²)

Let's simplify this step by step:

a = 800 / 400

a = 2 m/s²

Great! We've found the acceleration. The plane is accelerating at a rate of 2 meters per second squared. That means its speed is increasing by 2 meters per second every second. This is a crucial piece of information because now we can use it to find the final velocity – the speed of the plane at takeoff. Remember, we're using the concept of constant acceleration here, which makes these calculations possible.

Now that we have the acceleration, we are one step closer to finding our answer. Calculating the acceleration is often an intermediate step in physics problems, and it's important to understand what it represents in the context of the problem. In this case, it tells us how quickly the plane's speed is changing. So, with the acceleration in hand, let's move on to the final step: calculating the final velocity!

Calculating Final Velocity

Now that we know the acceleration (a = 2 m/s²), we can calculate the final velocity (v) using another kinematic equation. The equation that works best here is:

v = v₀ + at

Where:

v = final velocity (what we want to find)
v₀ = initial velocity (0 m/s, since the plane starts from rest)
a = acceleration (2 m/s²)
t = time (20 seconds)

Let's plug in the values:

v = 0 + (2 * 20)

Simplifying, we get:

v = 40 m/s

So, the final velocity of the plane at takeoff is 40 meters per second! That's our answer! We've successfully used the principles of kinematics to solve this problem. Remember, it's all about choosing the right equations and using the information you have to find what you need.

This result makes sense in the context of the problem. The plane starts from rest and accelerates for 20 seconds. An acceleration of 2 m/s² means the velocity increases by 2 m/s each second. After 20 seconds, the velocity would indeed be 40 m/s. So, we've not only solved the problem mathematically but also verified that the answer is reasonable. Next, we'll wrap up with a quick recap of the steps we took.

Final Answer and Conclusion

Okay, guys, we've cracked the code! The final velocity of the plane at takeoff is 40 m/s. That corresponds to answer choice (d).

Let's quickly recap the steps we took to solve this problem:

Understood the Problem: We identified the known values (time, distance, initial velocity) and the unknown value (final velocity).
Chose the Right Equation: We selected the appropriate kinematic equations to relate these variables.
Calculated Acceleration: We first used the equation d = v₀t + (1/2)at² to find the acceleration.
Calculated Final Velocity: Then, we used the equation v = v₀ + at to find the final velocity.
Verified the Answer: We checked that our answer made sense in the context of the problem.

This problem demonstrates how we can use basic physics principles and equations to solve real-world scenarios. Remember, the key is to break down the problem into smaller steps, identify the relevant information, and choose the right tools (equations) for the job. You got this, guys! Keep practicing, and you'll become a physics pro in no time!

If you have any questions or want to try another problem, let me know. Physics is all about understanding how the world works, and it's super cool when you can figure things out like this. So, great job today, and keep up the awesome work!