Sphere's Velocity After Upward Force: A Physics Problem

Let's break down this physics problem step-by-step, guys. We've got a sphere, initially chilling at rest, and then BAM! A constant force yanks it upwards. We need to figure out how fast it's moving after a couple of seconds, keeping gravity in mind. So, grab your thinking caps, and let's dive in!

Understanding the Forces at Play

First things first, let's identify all the forces acting on our sphere. We've got two main contenders here: the upward force applied and the ever-present force of gravity. The upward force is given as 30.0 N (Newtons). Gravity, on the other hand, always acts downwards and is calculated using the formula F = mg, where 'm' is the mass and 'g' is the acceleration due to gravity. In our case, m = 2.0 kg and g = 10 m/s², so the force of gravity is 2.0 kg * 10 m/s² = 20.0 N. It's crucial to recognize these opposing forces to understand the net force influencing the sphere's motion. Consider that the net force isn't simply the applied force, but the difference between the applied force and the gravitational force. When dealing with dynamics problems, always start by identifying all the forces and their directions. This meticulous approach will prevent common mistakes and lead to a clearer understanding of the physics involved. Moreover, correctly identifying the forces allows us to accurately calculate the net force, which is the key to determining the object's acceleration and subsequent motion.

Furthermore, take a moment to visualize the scenario. Imagine the sphere hanging in the air, and then the upward force begins to act upon it. The gravitational force is constantly pulling it down, while the applied force is trying to lift it up. The interplay between these forces determines whether the sphere accelerates upwards, downwards, or remains at rest. Understanding this dynamic interaction is fundamental to solving the problem. Additionally, keep in mind that forces are vector quantities, meaning they have both magnitude and direction. In this case, we have two forces acting along the same vertical line, but in opposite directions. Therefore, we can simply subtract their magnitudes to find the net force. This simplification makes the problem more manageable and allows us to focus on the essential aspects of the physics involved. Lastly, by understanding the forces at play, we can anticipate the sphere's motion and make informed predictions about its final velocity. This proactive approach enhances our problem-solving skills and deepens our understanding of Newtonian mechanics.

Calculating the Net Force and Acceleration

Alright, let's get calculating! We know the upward force and the force of gravity. To find the net force, we subtract the force of gravity from the upward force: 30.0 N - 20.0 N = 10.0 N. This means there's a net upward force of 10.0 N acting on the sphere. Now, with the net force in hand, we can determine the sphere's acceleration using Newton's second law of motion: F = ma, where 'F' is the net force, 'm' is the mass, and 'a' is the acceleration. Rearranging the formula to solve for acceleration, we get a = F/m. Plugging in our values, a = 10.0 N / 2.0 kg = 5.0 m/s². So, the sphere is accelerating upwards at a rate of 5.0 meters per second squared. Remember, understanding the relationship between force, mass, and acceleration is fundamental in physics. Newton's second law provides a powerful tool for analyzing the motion of objects under the influence of forces. By correctly applying this law, we can accurately predict how an object will accelerate in response to a given net force. Additionally, it's essential to pay attention to units. In this case, we're using Newtons for force, kilograms for mass, and meters per second squared for acceleration. Maintaining consistency in units ensures that our calculations are accurate and meaningful.

Furthermore, consider the implications of the calculated acceleration. The sphere is accelerating upwards, meaning its velocity is increasing over time. The magnitude of the acceleration tells us how quickly the velocity is changing. In this case, the velocity is increasing by 5.0 meters per second every second. This constant acceleration simplifies the problem, allowing us to use straightforward kinematic equations to determine the final velocity. Moreover, understanding the concept of acceleration is crucial for analyzing more complex scenarios involving varying forces and non-constant acceleration. By mastering the basics of Newtonian mechanics, we can tackle a wide range of physics problems and gain a deeper appreciation for the fundamental principles governing the motion of objects. Lastly, remember that acceleration is a vector quantity, meaning it has both magnitude and direction. In this case, the acceleration is directed upwards, indicating that the sphere is gaining speed in the upward direction.

Determining the Final Velocity

Now for the grand finale: finding the final velocity! We know the initial velocity (0 m/s since it starts at rest), the acceleration (5.0 m/s²), and the time (2.0 s). We can use the following kinematic equation: v = u + at, where 'v' is the final velocity, 'u' is the initial velocity, 'a' is the acceleration, and 't' is the time. Plugging in our values, v = 0 m/s + (5.0 m/s²)(2.0 s) = 10.0 m/s. Therefore, the intensity of the velocity of the sphere at the end of the 2.0-second interval is 10.0 m/s. This final calculation demonstrates the power of kinematic equations in predicting the motion of objects under constant acceleration. By carefully selecting the appropriate equation and plugging in the known values, we can accurately determine the final velocity of the sphere. Remember, kinematic equations are based on the assumption of constant acceleration. If the acceleration varies over time, more advanced techniques are required to analyze the motion. However, in this case, the constant upward force ensures that the acceleration remains constant, making the kinematic equation a perfect fit.

Furthermore, consider the significance of the final velocity. The sphere has accelerated from rest to a velocity of 10.0 m/s in just 2.0 seconds. This indicates the effectiveness of the applied force in overcoming gravity and accelerating the sphere upwards. The magnitude of the final velocity provides a quantitative measure of the sphere's motion and allows us to compare it to other scenarios. Moreover, understanding the relationship between initial velocity, acceleration, time, and final velocity is essential for solving a wide range of physics problems. By mastering these concepts, we can analyze the motion of objects in various contexts and gain a deeper understanding of the physical world. Lastly, remember that velocity is a vector quantity, meaning it has both magnitude and direction. In this case, the velocity is directed upwards, indicating that the sphere is moving upwards at a speed of 10.0 m/s.

Wrapping It Up

So, there you have it! By carefully analyzing the forces, calculating the net force and acceleration, and applying the appropriate kinematic equation, we successfully determined that the sphere's final velocity is 10.0 m/s. Physics problems like these might seem daunting at first, but breaking them down into smaller, manageable steps makes them much easier to tackle. Keep practicing, and you'll be a physics whiz in no time!