Calculating Car Acceleration And Distance A Physics Problem Solution
This article delves into a classic physics problem involving uniformly accelerated motion. We will explore how to determine the acceleration and distance covered by a car that starts from rest and reaches a certain velocity within a specified time frame. This problem serves as a fundamental example in understanding kinematics, the branch of physics that describes the motion of objects without considering the forces that cause the motion. By breaking down the problem step-by-step, we will reinforce the application of key kinematic equations and concepts. Mastering these principles is crucial for tackling more complex physics problems related to motion and dynamics. So, let's embark on this journey to unravel the mysteries of uniformly accelerated motion.
Problem Statement
A car starts from rest. After 5 seconds, its velocity becomes 30 m/s. Find the acceleration of the car and the distance covered by the car.
This problem presents a scenario of uniformly accelerated motion, where the car's velocity changes at a constant rate. To solve this, we'll use the fundamental equations of kinematics. The problem provides us with the initial velocity, final velocity, and the time interval, allowing us to calculate the acceleration and the distance traveled. Understanding these core concepts and applying them correctly will help us navigate similar physics problems in the future. The beauty of physics lies in its ability to predict and explain the world around us, and this problem provides a perfect example of how these principles work in action. So, let's dive into the solution and see how we can decipher the motion of this car.
Solution: Finding Acceleration
To determine the acceleration of the car, we will employ the first equation of motion, which establishes the relationship between final velocity (v), initial velocity (u), acceleration (a), and time (t):
v = u + at
In this problem:
- Initial velocity (u) = 0 m/s (since the car starts from rest)
- Final velocity (v) = 30 m/s
- Time (t) = 5 seconds
Substituting these values into the equation, we get:
- 30 m/s = 0 m/s + a * 5 s
Now, we solve for acceleration (a):
- 30 m/s = a * 5 s
- a = (30 m/s) / (5 s)
- a = 6 m/s²
Therefore, the acceleration of the car is 6 meters per second squared. This means that the car's velocity increases by 6 meters per second every second. This constant increase in velocity is the hallmark of uniformly accelerated motion. Understanding how to calculate acceleration from initial and final velocities, along with the time interval, is a crucial skill in physics. It allows us to analyze and predict the motion of objects in various scenarios. This result forms the basis for the next part of our problem, where we will calculate the distance covered by the car.
Solution: Finding Distance Covered
Now that we have determined the acceleration, we can calculate the distance (s) covered by the car using the second equation of motion:
s = ut + (1/2)at²
Where:
- s = distance
- u = initial velocity
- t = time
- a = acceleration
We already know:
- u = 0 m/s
- t = 5 s
- a = 6 m/s²
Plugging these values into the equation:
- s = (0 m/s)(5 s) + (1/2)(6 m/s²)(5 s)²
- s = 0 + (1/2)(6 m/s²)(25 s²)
- s = 3 m/s² * 25 s²
- s = 75 meters
Therefore, the car covers a distance of 75 meters in 5 seconds. This calculation demonstrates how the initial velocity, acceleration, and time all contribute to the total distance traveled. The equation we used encapsulates this relationship, allowing us to predict the distance an object will cover under uniform acceleration. This understanding is vital in many real-world applications, such as designing vehicles, analyzing projectile motion, and understanding the physics of collisions. By mastering these fundamental equations, we gain a powerful tool for understanding and predicting the motion of objects around us. This completes the solution to our problem, giving us both the acceleration and the distance covered by the car.
Alternative Method for Distance Calculation
Interestingly, there's another way to calculate the distance covered by the car. We can use the third equation of motion, which directly relates final velocity (v), initial velocity (u), acceleration (a), and distance (s):
v² = u² + 2as
We know:
- v = 30 m/s
- u = 0 m/s
- a = 6 m/s²
Substituting these values into the equation:
- (30 m/s)² = (0 m/s)² + 2 * (6 m/s²) * s
- 900 m²/s² = 0 + 12 m/s² * s
- s = (900 m²/s²) / (12 m/s²)
- s = 75 meters
As we can see, this method yields the same result as the previous one: the car covers a distance of 75 meters. This alternative approach highlights the interconnectedness of the kinematic equations. Depending on the information provided in a problem, one equation might be more convenient to use than another. The third equation of motion is particularly useful when we know the initial and final velocities and the acceleration, but not the time. By having multiple tools in our toolkit, we can tackle a wider range of physics problems with greater efficiency and confidence. This reinforces the importance of understanding the underlying principles and the relationships between different physical quantities.
Summary of Results
In summary, we have successfully determined the acceleration and distance covered by the car using the equations of motion. The results are:
- Acceleration of the car: 6 m/s²
- Distance covered by the car: 75 meters
These results provide a complete description of the car's motion during the 5-second interval. The acceleration tells us how the car's velocity is changing over time, and the distance tells us how far the car has traveled. This problem exemplifies the power of physics in describing and predicting the motion of objects. By applying the principles of kinematics and the equations of motion, we can analyze a wide variety of real-world scenarios, from the motion of cars and airplanes to the trajectory of projectiles. The ability to calculate acceleration and distance is fundamental to understanding more complex concepts in physics, such as work, energy, and momentum. This problem serves as a building block for further exploration of the fascinating world of physics.
Key Takeaways and Applications
This problem provides several key takeaways for understanding uniformly accelerated motion. First, it reinforces the importance of the equations of motion in relating displacement, velocity, acceleration, and time. These equations are the cornerstone of kinematics and are essential for solving a wide range of problems. Second, it highlights the concept of constant acceleration, where the velocity changes at a steady rate. This type of motion is common in many real-world scenarios, such as a car accelerating on a highway or a ball falling under the influence of gravity (neglecting air resistance). Third, it demonstrates that there can be multiple approaches to solving a problem. We calculated the distance using two different equations of motion, illustrating the flexibility and interconnectedness of physical principles.
The applications of these concepts extend far beyond textbook problems. In engineering, understanding motion is crucial for designing vehicles, machines, and structures. In sports, analyzing the motion of athletes and projectiles can improve performance. In everyday life, we use our intuitive understanding of motion to navigate our surroundings, drive cars, and play games. By mastering the fundamentals of kinematics, we gain a deeper appreciation for the physics that governs our world and develop the skills to solve real-world problems. This problem serves as a stepping stone for further exploration of more advanced topics in physics, such as dynamics, energy, and momentum.
Further Exploration
To further solidify your understanding of uniformly accelerated motion, consider exploring these additional avenues:
- Varying the initial conditions: Try solving the problem with different initial velocities. How does the acceleration and distance change if the car starts with a non-zero initial velocity?
- Introducing deceleration: What if the car is braking and slowing down? How would you adapt the equations of motion to handle negative acceleration (deceleration)?
- Graphical representation: Plot the car's velocity as a function of time. The slope of the line represents the acceleration. Plot the car's position as a function of time. What does the shape of the curve tell you about the motion?
- Real-world applications: Research how these concepts are used in real-world applications, such as designing vehicle safety systems or analyzing the motion of projectiles.
- More complex scenarios: Explore problems involving non-uniform acceleration or motion in two dimensions (projectile motion). These problems build upon the fundamental concepts we have discussed and provide a greater challenge.
By actively engaging with these concepts and exploring them in different contexts, you will develop a deeper understanding of physics and its applications. The journey of learning physics is one of continuous exploration and discovery, and this problem serves as a valuable starting point.
Conclusion
This exploration of a simple car acceleration problem has provided a valuable insight into the fundamental principles of kinematics. By applying the equations of motion, we were able to determine the car's acceleration and the distance it covered. We also saw how different equations can be used to solve the same problem, highlighting the interconnectedness of physical concepts. This problem serves as a foundation for understanding more complex scenarios involving motion. The key takeaways from this problem, including the equations of motion and the concept of constant acceleration, are essential tools for any student of physics. By understanding these principles, we can analyze and predict the motion of objects in a wide range of situations. The study of physics is about understanding the fundamental laws that govern the universe, and this problem provides a glimpse into the power and elegance of these laws. By continuing to explore and apply these concepts, we can deepen our understanding of the world around us and unlock new possibilities for innovation and discovery.