Calculating Total Distance With Acceleration And Braking

Oct 20, 2025 by SLV Team 57 views

Let's dive into the fascinating world of motion and calculations! Today, we're going to break down a problem involving a mobile object that starts from rest, accelerates, and then brakes to a halt. We'll be calculating the total distance traveled, which is a classic physics problem that combines concepts of uniformly accelerated motion. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's clearly understand the scenario. We have a mobile object (think of it as a car, a scooter, or even a futuristic hovercraft!) that begins its journey from a standstill. It then accelerates at a constant rate for a specific time, after which it applies brakes and decelerates until it comes to a complete stop. Our mission is to find the total distance covered during this entire process. This problem involves two distinct phases of motion: acceleration and deceleration, each with its own set of parameters.

To solve this, we'll need to break down the problem into smaller, manageable parts. We'll first analyze the acceleration phase, then the deceleration phase, and finally, we'll add the distances covered in each phase to find the total distance. Remember, physics problems are often like puzzles; breaking them down into smaller pieces makes them much easier to solve. So, let's put on our detective hats and start piecing this puzzle together!

Phase 1: Acceleration

In the initial phase, the mobile starts from rest, meaning its initial velocity is zero. It then accelerates at a rate of 4 m/s² for a duration of 10 seconds. This is uniformly accelerated motion (M.U.A), which means the velocity increases at a constant rate. To find the distance covered during this phase, we can use the following equation of motion:

d = v₀t + (1/2)at²

Where:

d is the distance traveled
v₀ is the initial velocity (0 m/s in this case)
t is the time (10 s)
a is the acceleration (4 m/s²)

Plugging in the values, we get:

d = (0 m/s)(10 s) + (1/2)(4 m/s²)(10 s)²
d = 0 + (2 m/s²)(100 s²)
d = 200 meters

So, during the acceleration phase, the mobile covers a distance of 200 meters. But we're not done yet! We still need to figure out the distance covered during braking.

Phase 2: Deceleration (Braking)

Now comes the braking phase. The mobile, which has gained speed during acceleration, now applies brakes and decelerates. This means its velocity decreases at a constant rate. The problem states that the mobile decelerates at 6 m/s² for 20 seconds until it stops. It's important to note that deceleration is just acceleration in the opposite direction, so we'll treat it as a negative acceleration (-6 m/s²).

To find the distance covered during braking, we'll use the same equation of motion as before:

d = v₀t + (1/2)at²

However, there's a slight twist! We need to figure out the initial velocity (v₀) for this phase. The initial velocity during braking is the final velocity the mobile attained during acceleration. So, let's calculate that first. We can use another equation of motion:

v = v₀ + at

Where:

v is the final velocity
v₀ is the initial velocity (0 m/s for the acceleration phase)
a is the acceleration (4 m/s²)
t is the time (10 s)

Plugging in the values, we get:

v = 0 m/s + (4 m/s²)(10 s)
v = 40 m/s

So, the final velocity after the acceleration phase is 40 m/s. This becomes the initial velocity for the braking phase. Now we can calculate the distance covered during braking:

d = v₀t + (1/2)at²
d = (40 m/s)(20 s) + (1/2)(-6 m/s²)(20 s)²
d = 800 m + (-3 m/s²)(400 s²)
d = 800 m - 1200 m
d = -400 meters

Wait a minute! A negative distance? That doesn't make sense, does it? This is a crucial point! The equation we used assumes constant acceleration over the entire time period. However, in this case, the mobile stops before the 20 seconds have elapsed. We've calculated the distance the mobile would have traveled if it continued decelerating for the full 20 seconds, but that's not what actually happened. We need to find the actual time it takes to stop.

To find the actual time to stop, we use the equation:

v = v₀ + at

Where v = 0 m/s (final velocity, as the mobile stops).

0 m/s = 40 m/s + (-6 m/s²)t
6 m/s² * t = 40 m/s
t = 40 m/s / 6 m/s²
t ≈ 6.67 seconds

Now that we have the correct time, we can calculate the distance covered during braking:

d = v₀t + (1/2)at²
d = (40 m/s)(6.67 s) + (1/2)(-6 m/s²)(6.67 s)²
d ≈ 266.8 m - 133.4 m
d ≈ 133.4 meters

So, the mobile covers approximately 133.4 meters during braking. See how important it is to think critically about the results and not just blindly apply formulas? Always check if your answers make sense in the context of the problem!

Calculating the Total Distance

Now that we know the distances covered during both phases, calculating the total distance is a piece of cake! We simply add the distances together:

Total distance = Distance during acceleration + Distance during braking
Total distance = 200 meters + 133.4 meters
Total distance ≈ 333.4 meters

So, the total distance traveled by the mobile is approximately 333.4 meters. Woohoo! We solved it!

Key Concepts and Takeaways

This problem beautifully illustrates the application of uniformly accelerated motion concepts. Here are some key takeaways:

Understanding the Problem: Always start by carefully reading and understanding the problem statement. Identify the knowns (given information) and the unknowns (what you need to find).
Breaking Down the Problem: Complex problems can be simplified by breaking them down into smaller, manageable steps. In this case, we separated the motion into acceleration and deceleration phases.
Equations of Motion: Remember the equations of motion for uniformly accelerated motion. These are your tools for solving these types of problems.
Critical Thinking: Don't just plug in numbers and hope for the best. Think critically about the results and ensure they make sense in the context of the problem. Negative distances or times often indicate an error or a need to adjust your approach.
Units: Always pay attention to units and ensure they are consistent throughout your calculations.

Real-World Applications

The concepts we've used in this problem are not just theoretical; they have numerous real-world applications. Understanding motion, acceleration, and braking distances is crucial in:

Vehicle Design: Engineers use these principles to design safer and more efficient vehicles.
Traffic Engineering: Traffic flow, traffic light timing, and road design all rely on understanding motion and braking distances.
Sports: Analyzing the motion of athletes and projectiles in sports like baseball, basketball, and track and field.
Robotics: Programming robots to move accurately and efficiently requires a solid understanding of motion control.

Conclusion

Guys, we've successfully navigated through a problem involving acceleration, braking, and distance calculations! By breaking down the problem, applying the equations of motion, and thinking critically about our results, we were able to find the total distance traveled by the mobile. Remember, physics is all about understanding the world around us, and problems like this help us develop those understanding skills. So, keep practicing, keep exploring, and keep those brain gears turning! You've got this! And hey, who knows, maybe you'll be designing the next generation of self-driving cars or figuring out how to land a spacecraft on Mars!

Remember, learning physics is like building a house. You start with the basics, lay the foundation, and gradually build upon it. Each problem you solve is like adding another brick to your wall of knowledge. So, don't get discouraged if things seem challenging at first. Keep practicing, keep asking questions, and keep building that amazing physics house of yours! You'll be amazed at what you can achieve!

So, that's it for this problem. I hope you found this explanation helpful and insightful. If you have any more questions or want to explore other physics concepts, don't hesitate to ask. And remember, physics is not just about formulas and equations; it's about understanding the world around you in a deeper and more meaningful way. So, go out there and explore the physics of everyday life! You might be surprised at what you discover. Until next time, keep learning, keep exploring, and keep having fun with physics!