Projectile Motion Problem Solving Velocity After Gravity Disappears

This article delves into a classic physics problem involving projectile motion and the intriguing scenario where gravity suddenly disappears. We will dissect the problem step-by-step, applying fundamental physics principles to arrive at the solution. This exploration will not only help in understanding the specific question but also reinforce core concepts like initial velocity, acceleration due to gravity, and inertia. Let's embark on this journey of understanding projectile motion and inertia.

Problem Statement: Decoding the Vertical Ascent

The problem presents a scenario where a body is projected vertically upwards with an initial velocity of 98 m/s. After 2 seconds, a hypothetical situation occurs: the acceleration due to gravity of Earth vanishes instantaneously. Our mission is to determine the velocity of the body at the end of the subsequent 3 seconds. This question challenges our understanding of how gravity influences motion and what happens when that influence is suddenly removed. The interplay between the upward motion imparted by the initial velocity and the downward pull of gravity is crucial to grasp. The sudden disappearance of gravity introduces the concept of inertia, which dictates that an object in motion will continue in motion with the same velocity unless acted upon by an external force. This is a key element in solving the problem.

Breaking Down the Initial Phase: Upward Trajectory Under Gravity

Before gravity disappears, the body experiences constant deceleration due to gravity (approximately 9.8 m/s²). To understand its motion, we need to calculate the body's velocity after the initial 2 seconds. We can use the first equation of motion: v = u + at, where:

• v is the final velocity
• u is the initial velocity (98 m/s)
• a is the acceleration (-9.8 m/s², negative as it opposes the upward motion)
• t is the time (2 s)

Plugging in the values, we get:

v = 98 m/s + (-9.8 m/s²) * 2 s v = 98 m/s - 19.6 m/s v = 78.4 m/s

This calculation reveals that after 2 seconds, the body's upward velocity has decreased to 78.4 m/s due to the constant deceleration caused by gravity. This is a critical piece of information, as it becomes the initial velocity for the next phase of the motion when gravity vanishes. This phase highlights the importance of understanding the cumulative effect of gravity on a projectile's velocity over time. The initial velocity significantly impacts the subsequent motion, and calculating this intermediate velocity is essential for accurately predicting the body's behavior.

The Moment Gravity Vanishes: Embracing Inertia

Now comes the pivotal moment: gravity disappears. This means the body is no longer subject to gravitational acceleration. Here, Newton's First Law of Motion, the law of inertia, takes center stage. Inertia dictates that an object in motion will stay in motion with the same velocity unless acted upon by an external force. Since gravity, the decelerating force, has vanished, the body will continue moving upwards with the velocity it had at the instant gravity disappeared, which we calculated to be 78.4 m/s. This is a crucial concept to grasp. Many might intuitively think the body will stop or change direction, but inertia ensures that the motion continues unchanged in the absence of external forces. For the next 3 seconds, the body will maintain this constant upward velocity. This highlights a fundamental difference between motion under gravity and motion in the absence of gravity.

Calculating the Final Velocity: Inertia in Action

Since the body is moving with a constant velocity of 78.4 m/s for the next 3 seconds (because there's no acceleration), its velocity at the end of these 3 seconds will remain the same. There's no force to change its speed or direction. This directly follows the principle of inertia. Therefore, the final velocity of the body after the next 3 seconds is 78.4 m/s. This seemingly simple calculation underscores the power of inertia. It also contrasts sharply with projectile motion under gravity, where velocity constantly changes due to gravitational acceleration. The absence of gravity simplifies the problem significantly, as we don't need to consider changing velocity. The final velocity is simply the velocity at the instant gravity vanished. This step reinforces the concept of constant velocity motion and its applicability when external forces are absent.

Solution and Answer

Therefore, the velocity of the body at the end of the next 3 seconds is 78.4 m/s. This corresponds to option 3 in the given choices. This solution beautifully illustrates the combined effect of projectile motion under gravity and the principle of inertia. We first calculated the velocity after 2 seconds of upward motion against gravity, and then we applied the concept of inertia to determine the velocity when gravity disappeared. Understanding these concepts is vital for solving a wide range of physics problems related to motion and forces. This problem also highlights the importance of carefully analyzing the problem statement and identifying the key physical principles involved. By breaking down the problem into smaller, manageable steps, we can arrive at the solution logically and systematically. The answer not only provides a numerical value but also deepens our understanding of the underlying physics.

Understanding the Fundamentals of Projectile Motion

Projectile motion is a cornerstone of classical mechanics, describing the motion of an object projected into the air, subject only to the acceleration of gravity. Understanding projectile motion is essential for grasping a wide array of physical phenomena, from the trajectory of a thrown ball to the flight of a rocket. The key to analyzing projectile motion lies in recognizing that it can be broken down into two independent components: horizontal and vertical motion. The vertical motion is influenced by gravity, causing the object to accelerate downwards, while the horizontal motion, in the absence of air resistance, remains constant. This independence allows us to treat each component separately, simplifying the analysis and making it easier to predict the projectile's path. To truly grasp projectile motion, we must understand concepts like initial velocity, launch angle, and range. The initial velocity dictates how fast the object is launched, and the launch angle determines the trajectory's shape. The range, the horizontal distance traveled by the projectile, depends on both the initial velocity and the launch angle. Furthermore, understanding the effect of gravity, a constant downward acceleration, is crucial. The acceleration due to gravity constantly changes the vertical velocity of the projectile, causing it to slow down as it ascends and speed up as it descends. This interplay between the initial conditions and gravity shapes the parabolic path characteristic of projectile motion. Learning to apply these principles allows us to solve problems involving various scenarios, such as calculating the time of flight, the maximum height reached, or the horizontal range. In essence, mastering projectile motion provides a solid foundation for understanding more advanced topics in physics.

The Role of Gravity in Shaping Trajectory

In the realm of projectile motion, gravity plays a pivotal role in shaping the trajectory of an object. The constant downward force exerted by gravity is the primary factor influencing the vertical component of the projectile's motion. This influence manifests as a constant downward acceleration, approximately 9.8 m/s², causing the object to slow down as it moves upwards and accelerate as it falls back down. Gravity's effect on trajectory is significant, dictating the shape and characteristics of the parabolic path typically observed in projectile motion. To illustrate, consider an object launched vertically upwards. Its initial velocity is directed upwards, but gravity acts in the opposite direction, causing the object to decelerate until it momentarily comes to rest at its highest point. Then, gravity pulls it back down, accelerating it until it reaches its initial launch height. This up-and-down motion is entirely governed by gravity. When an object is launched at an angle, the same principles apply, but the vertical motion is coupled with a constant horizontal motion. The parabolic trajectory is a result of the continuous interplay between the initial velocity, the launch angle, and the constant acceleration due to gravity. The range of the projectile, the horizontal distance it covers, is also significantly affected by gravity. A stronger gravitational force would result in a shorter range, while a weaker gravitational force would lead to a longer range, assuming the initial velocity and launch angle remain constant. Therefore, a comprehensive understanding of gravity's role is indispensable for accurately predicting and analyzing projectile motion. This understanding allows us to solve problems related to trajectory, range, time of flight, and maximum height reached by a projectile. Furthermore, it forms the basis for understanding more complex scenarios, such as the motion of projectiles in the presence of air resistance or other external forces.

Inertia: The Unseen Force Maintaining Motion

Inertia, a fundamental concept in physics, often goes unnoticed but plays a crucial role in maintaining motion. Inertia, defined as the tendency of an object to resist changes in its state of motion, is a core principle articulated in Newton's First Law of Motion. This law states that an object at rest will stay at rest, and an object in motion will stay in motion with the same velocity, unless acted upon by an external force. Inertia is not a force itself but rather a property of matter that resists acceleration. The more massive an object is, the greater its inertia, meaning it requires a greater force to change its velocity. This principle is readily observable in everyday life. For example, a car requires significant force to start moving because of its inertia, and once it's moving, it tends to maintain its motion unless the brakes are applied or friction slows it down. Similarly, a heavy box is harder to push than a light box because the heavy box possesses greater inertia. In the context of projectile motion, inertia is responsible for maintaining the horizontal component of the projectile's velocity. In the absence of air resistance, there is no horizontal force acting on the projectile, so its horizontal velocity remains constant due to inertia. This constant horizontal motion, combined with the vertical motion influenced by gravity, gives rise to the characteristic parabolic trajectory. Understanding inertia is essential for comprehending how objects move and interact in the world around us. It helps us explain why objects resist changes in their motion, why massive objects are harder to move, and how motion is maintained in the absence of external forces. Furthermore, it serves as a foundation for understanding other concepts in physics, such as momentum and collisions. Mastering inertia allows us to predict and explain a wide range of phenomena, from the motion of everyday objects to the behavior of celestial bodies.

Applying Physics Principles to Solve the Problem

To effectively solve the problem of a body projected vertically upwards with disappearing gravity, a systematic application of physics principles is crucial. The problem requires us to consider the motion in two distinct phases: the motion under gravity before gravity disappears and the motion under inertia after gravity vanishes. In the first phase, the body is subject to constant deceleration due to gravity. To analyze this phase, we can employ the equations of motion, which relate displacement, velocity, acceleration, and time. Specifically, the equation v = u + at is particularly useful for determining the velocity of the body after a certain time, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. Once we determine the velocity at the instant gravity disappears, we transition to the second phase. Here, the principle of inertia takes precedence. Since there is no external force acting on the body (gravity has vanished), it will continue to move with the same velocity it had at the end of the first phase. This means the velocity remains constant during the subsequent time interval. Therefore, the final velocity is simply equal to the velocity calculated at the end of the first phase. Applying these principles systematically allows us to break down the problem into manageable steps and arrive at the solution logically. This approach not only provides the answer but also reinforces our understanding of the underlying physics. Furthermore, it highlights the importance of identifying the relevant concepts and principles for each phase of the motion. By mastering this problem-solving approach, we can tackle a wide range of physics problems involving motion, forces, and inertia. The key is to carefully analyze the problem statement, identify the key physical principles involved, and apply them in a step-by-step manner. This systematic approach fosters a deeper understanding of physics and enhances our ability to solve complex problems.

Step-by-Step Solution Breakdown

Solving physics problems effectively requires a structured approach, and this problem of a body projected upwards with disappearing gravity is no exception. A step-by-step solution breakdown ensures clarity and accuracy in the process. The first step is to analyze the initial motion under gravity. This involves identifying the known quantities: the initial velocity (98 m/s), the acceleration due to gravity (-9.8 m/s²), and the time duration (2 seconds). The objective is to determine the velocity of the body at the end of these 2 seconds. We can use the first equation of motion, v = u + at, to calculate this velocity. Substituting the known values, we get v = 98 m/s + (-9.8 m/s²) * 2 s = 78.4 m/s. This calculated velocity becomes the initial velocity for the next phase of the motion. The second step is to consider the motion after gravity disappears. According to the principle of inertia, the body will continue moving with the same velocity it had at the instant gravity vanished, which is 78.4 m/s. Since there is no external force acting on the body, the velocity remains constant for the next 3 seconds. Therefore, the final velocity of the body after 3 seconds is 78.4 m/s. This concludes the solution. By following this structured step-by-step approach, we not only arrive at the correct answer but also gain a deeper understanding of the problem and the underlying physics principles. This method is applicable to a wide range of physics problems and promotes a systematic and logical problem-solving mindset. Furthermore, it allows us to identify potential errors and correct them more easily. The key is to break down the problem into smaller, manageable steps, solve each step individually, and then combine the results to obtain the final solution. This process enhances our problem-solving skills and reinforces our understanding of physics concepts.

Importance of Understanding Conceptual Physics

While mathematical calculations are essential in physics, a strong understanding of conceptual physics is equally vital for problem-solving and a deeper appreciation of the subject. Conceptual physics focuses on the underlying principles and concepts that govern physical phenomena, rather than just the mathematical formulas. It involves understanding why things happen the way they do, not just how to calculate them. In the context of this problem, conceptual understanding allows us to recognize that the disappearance of gravity invokes the principle of inertia, which dictates that the body will continue moving at a constant velocity in the absence of external forces. Without this conceptual understanding, one might incorrectly assume that the body will stop or change direction. Conceptual physics also helps us to interpret the results of calculations and determine if they make sense in the real world. For example, knowing that gravity causes objects to accelerate downwards helps us understand why the velocity of the body decreases in the first phase of the motion. Similarly, understanding that inertia maintains constant velocity in the absence of forces helps us to predict the body's motion in the second phase. Conceptual understanding enhances our ability to solve problems by allowing us to choose the appropriate formulas, interpret the results correctly, and avoid common mistakes. It also fosters a deeper appreciation for the beauty and elegance of physics. By combining conceptual understanding with mathematical skills, we can effectively tackle a wide range of physics problems and develop a more comprehensive understanding of the physical world. The key is to focus not only on memorizing formulas but also on understanding the underlying principles and concepts. This approach will lead to greater success in physics and a more profound appreciation for the subject.

Conclusion: Mastering Motion and Gravity

In conclusion, solving this problem of a body projected vertically upwards with disappearing gravity has provided valuable insights into the principles of motion, gravity, and inertia. We have seen how the systematic application of physics principles, combined with a strong conceptual understanding, can lead to a clear and accurate solution. By breaking down the problem into manageable steps and carefully analyzing each phase of the motion, we were able to determine the final velocity of the body. This problem serves as a reminder of the importance of understanding both the mathematical and conceptual aspects of physics. While the equations of motion provide the tools for calculation, conceptual understanding guides us in choosing the appropriate equations and interpreting the results correctly. Furthermore, it highlights the interconnectedness of different physics concepts. The principle of inertia, which seems simple on the surface, plays a crucial role in understanding motion in the absence of external forces. Gravity, a fundamental force, governs the vertical motion of projectiles. By mastering these concepts, we can confidently tackle a wide range of physics problems related to motion and forces. The key takeaway is that physics is not just about memorizing formulas; it's about understanding the underlying principles and applying them logically and systematically. This approach will not only lead to success in physics but also foster a deeper appreciation for the physical world around us. The journey of learning physics is a continuous one, and each problem solved adds to our understanding and strengthens our problem-solving skills. This particular problem, with its intriguing scenario of disappearing gravity, provides a unique opportunity to reinforce our grasp of fundamental concepts and develop a more intuitive understanding of motion and forces.