Solving Displacement With Uniform Acceleration A Physics Guide
In the realm of physics, understanding the motion of objects is fundamental. This article delves into a classic problem involving a body initially at rest, undergoing uniform acceleration, and subsequently moving at a constant velocity. We will dissect the problem, break down the concepts, and arrive at the solution, providing a comprehensive understanding of displacement calculations in such scenarios.
Understanding the Problem
To effectively tackle this problem, it's crucial to first grasp the core concepts involved. Displacement refers to the change in position of an object, essentially how far it has moved from its starting point. Uniform acceleration signifies that the velocity of the object changes at a constant rate over time. In this particular case, the body begins at rest, meaning its initial velocity is zero. It then accelerates uniformly at a rate of 10 m/s² for the first 10 seconds. This implies that its velocity increases by 10 meters per second every second. Following this acceleration phase, the body maintains the maximum velocity it attained during the acceleration phase for another 10 seconds. The ultimate goal is to determine the total displacement of the body throughout this entire 20-second journey.
Breaking Down the Motion
The motion of the body can be divided into two distinct phases: the acceleration phase and the constant velocity phase. During the acceleration phase, the body's velocity is constantly changing due to the uniform acceleration. To calculate the displacement during this phase, we can employ the following kinematic equation:
s = ut + (1/2)at²
where:
- s represents the displacement
- u denotes the initial velocity
- t signifies the time elapsed
- a represents the acceleration
In our scenario, u = 0 m/s (since the body starts from rest), a = 10 m/s², and t = 10 s. Plugging these values into the equation, we get:
s = (0)(10) + (1/2)(10)(10)² = 500 meters
Therefore, the displacement during the acceleration phase is 500 meters.
Next, we need to determine the displacement during the constant velocity phase. To do this, we first need to calculate the maximum velocity attained by the body during the acceleration phase. We can use the following equation:
v = u + at
where:
- v represents the final velocity
In this case, v = 0 + (10)(10) = 100 m/s. This means the body reaches a maximum velocity of 100 meters per second. During the constant velocity phase, the body maintains this velocity for 10 seconds. The displacement during this phase can be calculated using the simple formula:
s = vt
where:
- s represents the displacement
- v denotes the velocity
- t signifies the time elapsed
In our scenario, v = 100 m/s and t = 10 s. Therefore, s = (100)(10) = 1000 meters. The displacement during the constant velocity phase is 1000 meters.
Calculating the Total Displacement
To find the total displacement, we simply add the displacements from the two phases:
Total displacement = Displacement during acceleration phase + Displacement during constant velocity phase
Total displacement = 500 meters + 1000 meters = 1500 meters
Therefore, the total displacement of the body is 1500 meters.
Conclusion
This problem exemplifies the application of fundamental kinematic equations to analyze motion with uniform acceleration and constant velocity. By breaking down the motion into distinct phases and applying the appropriate equations, we can accurately calculate the displacement of the body. The key takeaway is the importance of understanding the concepts of displacement, uniform acceleration, and constant velocity, and how they relate to each other in describing the motion of objects. This problem provides a solid foundation for tackling more complex scenarios in kinematics and dynamics. Understanding these concepts is crucial for comprehending the world around us, from the motion of vehicles to the trajectory of projectiles. By mastering these principles, we unlock a deeper understanding of the physical laws that govern our universe.
To effectively solve physics problems, a structured approach is essential. Let's outline a step-by-step solution to the problem we've been discussing, highlighting key strategies that can be applied to a wide range of physics problems.
1. Understand the Problem Statement:
The first and most crucial step is to thoroughly understand the problem statement. Read the problem carefully, identifying the known quantities, the unknown quantity you need to find, and any specific conditions or constraints. In our case, we know the initial velocity (0 m/s), the acceleration (10 m/s²), the time for acceleration (10 s), and the time for constant velocity motion (10 s). We need to find the total displacement. It's often helpful to visualize the scenario, perhaps sketching a simple diagram to represent the motion.
2. Identify Relevant Concepts and Equations:
Next, identify the relevant physics concepts and equations that apply to the problem. In this scenario, we are dealing with kinematics, specifically motion with uniform acceleration and constant velocity. The key equations we need are:
- Equation for displacement with uniform acceleration: s = ut + (1/2)at²
- Equation for final velocity with uniform acceleration: v = u + at
- Equation for displacement with constant velocity: s = vt
Recognizing these equations is crucial for setting up the solution.
3. Break Down the Problem into Stages:
Complex problems are often easier to solve when broken down into smaller, manageable stages. In this case, we identified two distinct phases of motion: the acceleration phase and the constant velocity phase. Solve for the unknowns in each phase separately. This simplifies the calculations and reduces the chance of errors.
4. Solve for Unknowns in Each Stage:
Now, apply the relevant equations to each stage, substituting the known values and solving for the unknowns. In the acceleration phase, we used s = ut + (1/2)at² to find the displacement and v = u + at to find the final velocity (which becomes the constant velocity in the next phase). In the constant velocity phase, we used s = vt to find the displacement.
5. Combine Results from Each Stage:
Once you've solved for the unknowns in each stage, combine the results to answer the overall problem. In this case, we added the displacements from the acceleration phase and the constant velocity phase to find the total displacement.
6. Check Your Answer:
Finally, always check your answer for reasonableness. Does the magnitude of the displacement make sense given the acceleration and time intervals? Are the units correct? A quick sanity check can help identify potential errors.
At the heart of this problem lie three fundamental concepts in physics: displacement, velocity, and acceleration. A deep understanding of these concepts is essential not only for solving physics problems but also for comprehending the motion of objects in the real world. Let's delve into each of these concepts in more detail.
Displacement: The Change in Position
Displacement is a vector quantity that refers to the change in position of an object. It's not simply the total distance traveled; rather, it's the shortest distance between the initial and final positions, along with the direction. For instance, if an object moves 5 meters to the right and then 2 meters to the left, the total distance traveled is 7 meters, but the displacement is 3 meters to the right. Displacement is a crucial concept because it provides information about the net change in position, which is often more important than the total path traveled. In our problem, we calculated the displacement in each phase of motion and then added them together to find the total displacement.
Velocity: The Rate of Change of Displacement
Velocity is another vector quantity that describes the rate at which an object's displacement changes over time. It essentially tells us how fast an object is moving and in what direction. Velocity is calculated as displacement divided by time. Like displacement, velocity has both magnitude (speed) and direction. The unit of velocity is typically meters per second (m/s). In our problem, we calculated the final velocity after the acceleration phase, which then became the constant velocity during the second phase of motion. Understanding the relationship between displacement and velocity is crucial for analyzing motion.
Acceleration: The Rate of Change of Velocity
Acceleration is the rate at which an object's velocity changes over time. It's also a vector quantity, meaning it has both magnitude and direction. Acceleration can be positive (increasing velocity), negative (decreasing velocity, also called deceleration), or zero (constant velocity). The unit of acceleration is typically meters per second squared (m/s²). In our problem, the body experienced uniform acceleration, meaning its velocity changed at a constant rate. Acceleration is the key factor that causes changes in motion, and understanding it is vital for analyzing how objects move under the influence of forces.
The Interplay of Displacement, Velocity, and Acceleration
These three concepts are intricately linked. Velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. Understanding these relationships allows us to predict and analyze the motion of objects in a wide range of scenarios. For example, if we know the acceleration of an object and its initial velocity, we can calculate its velocity and displacement at any time in the future. The kinematic equations we used in this problem are mathematical expressions of these relationships.
The concepts explored in this problem, such as displacement, velocity, and acceleration, are not confined to textbooks and theoretical exercises. They are fundamental to understanding and analyzing motion in countless real-world scenarios. Let's explore some practical applications of these concepts.
Transportation:
The motion of vehicles, whether cars, trains, or airplanes, is governed by the principles of kinematics. The acceleration of a car, for instance, determines how quickly it can reach a certain velocity. The displacement of a train over a given time is crucial for scheduling and logistics. Air traffic controllers use these concepts to ensure the safe separation of aircraft, taking into account their velocities and trajectories. Understanding the relationship between acceleration, velocity, and displacement is essential for designing safer and more efficient transportation systems. For example, the design of braking systems in cars relies heavily on these principles to ensure that vehicles can decelerate safely and effectively.
Sports:
Many sports involve objects in motion, and understanding kinematics is crucial for analyzing and optimizing athletic performance. The trajectory of a baseball thrown by a pitcher, the jump of a basketball player, and the motion of a golf ball after being struck are all governed by the laws of physics. Athletes and coaches use these concepts to improve technique, predict outcomes, and develop strategies. For example, analyzing the velocity and angle of release of a javelin can help athletes maximize their throwing distance. Similarly, understanding the acceleration and deceleration phases of a sprint can help runners optimize their running form and speed.
Engineering:
Engineers across various disciplines rely heavily on kinematics and dynamics. Civil engineers use these principles to design bridges and buildings that can withstand various forces and stresses. Mechanical engineers apply them to design machines and engines, optimizing their performance and efficiency. Aerospace engineers use them to design aircraft and spacecraft, ensuring their stability and maneuverability. For example, the design of a roller coaster involves careful calculations of acceleration, velocity, and displacement to ensure a thrilling yet safe ride. Similarly, the design of a robotic arm requires precise control of its motion, which relies on a thorough understanding of kinematics.
Everyday Life:
Even in our daily lives, we encounter these concepts regularly. When we walk, run, or cycle, we are experiencing displacement, velocity, and acceleration. Understanding these concepts can help us make informed decisions, such as estimating travel times or judging the speed of oncoming traffic. For example, when crossing a street, we instinctively assess the velocity of approaching cars and their distance (related to displacement) to determine if it's safe to cross. The ability to intuitively understand and apply these concepts is a testament to their fundamental nature in our world.
In conclusion, the problem of a body starting from rest and moving with uniform acceleration provides a valuable framework for understanding fundamental concepts in physics. By breaking down the problem into stages, applying the appropriate kinematic equations, and carefully analyzing the results, we can arrive at the solution and gain a deeper appreciation for the principles of motion. Mastering these fundamentals is not only essential for success in physics but also for understanding the world around us and making informed decisions in various aspects of our lives. From transportation and sports to engineering and everyday activities, the concepts of displacement, velocity, and acceleration are interwoven into the fabric of our physical reality. By continuing to explore and apply these principles, we can unlock a greater understanding of the universe and our place within it.
The correct answer is 3) 1500 m.
