Truck Final Velocity Calculation: A Physics Problem
Hey physics enthusiasts! Ever wondered how to calculate the final velocity of a truck given its initial conditions and the force applied? Let's dive into a fascinating problem where we'll explore the physics behind a moving truck. We have a scenario where a 412 kg truck is cruising on a flat surface, initially moving at 13.0 m/s. The truck's engine kicks in, generating a 1005 N forward force over a distance of 9.85 m. Our mission? To determine the truck's final velocity. This problem combines concepts from kinematics and dynamics, requiring us to use Newton's laws of motion and work-energy principles. So, buckle up, and let's get started!
Understanding the Problem
Before we jump into calculations, let's break down the problem. We need to identify the knowns and unknowns, and then strategize how to connect them using physics principles. The knowns are: the truck's mass (m = 412 kg), initial velocity (Vi = 13.0 m/s), the applied force (F = 1005 N), and the distance over which the force is applied (d = 9.85 m). The unknown is the final velocity (Vf), which is what we are trying to find. We're dealing with motion under constant force, which means the truck is accelerating. This hints at using equations of motion that relate initial velocity, final velocity, acceleration, and distance. Additionally, the force applied over a distance suggests the involvement of work and energy principles. Understanding these connections is crucial for solving the problem effectively. By carefully considering the given information and the physics concepts involved, we can map out a clear path to finding the final velocity of the truck. This initial step of problem analysis is essential in physics, as it sets the stage for accurate and efficient problem-solving. Now that we have a solid grasp of what we know and what we need to find, let's move on to the next step: selecting the appropriate physics principles and equations to solve this problem.
Applying Physics Principles
To solve this problem, we will primarily use two fundamental physics concepts: Newton's Second Law of Motion and the Work-Energy Theorem. Newton's Second Law (F = ma) tells us that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This law will help us determine the acceleration of the truck given the applied force and the truck's mass. The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. Mathematically, it's expressed as W = ΔKE, where W is the work done, and ΔKE is the change in kinetic energy. The work done can be calculated as W = Fd, where F is the force applied, and d is the distance over which the force is applied. The kinetic energy (KE) is given by the formula KE = (1/2)mv^2, where m is the mass, and v is the velocity. By combining these principles, we can relate the force applied to the change in the truck's velocity. First, we'll use Newton's Second Law to find the acceleration. Then, we'll use the Work-Energy Theorem to connect the work done by the engine to the change in kinetic energy, which will allow us to calculate the final velocity. This step-by-step approach ensures that we use the fundamental principles of physics to accurately solve the problem, giving us a clear and logical path to the final answer. Now that we have our physics toolkit ready, let's start crunching the numbers!
Step-by-Step Solution
Let's break down the solution into manageable steps.
Step 1: Calculate the Acceleration
Using Newton's Second Law (F = ma), we can find the acceleration (a) of the truck. We know the force (F = 1005 N) and the mass (m = 412 kg). Plugging these values into the equation, we get:
1005 N = 412 kg * a
Solving for a:
a = 1005 N / 412 kg ≈ 2.44 m/s²
So, the truck is accelerating at approximately 2.44 meters per second squared. This value represents how quickly the truck's velocity is changing due to the engine's force.
Step 2: Apply the Work-Energy Theorem
The Work-Energy Theorem (W = ΔKE) states that the work done on an object equals its change in kinetic energy. The work done (W) is the force (F) times the distance (d), which is:
W = Fd = 1005 N * 9.85 m ≈ 9899.25 J
The change in kinetic energy (ΔKE) is the final kinetic energy (KEf) minus the initial kinetic energy (KEi):
ΔKE = KEf - KEi = (1/2)mVf² - (1/2)mVi²
Plugging in the values we know:
ΔKE = (1/2)(412 kg)Vf² - (1/2)(412 kg)(13.0 m/s)²
Since W = ΔKE, we can set these two expressions equal to each other:
9899.25 J = (1/2)(412 kg)Vf² - (1/2)(412 kg)(13.0 m/s)²
Now, let's simplify and solve for the final velocity (Vf).
Step 3: Solve for the Final Velocity (Vf)
From the previous step, we have the equation:
9899.25 J = (1/2)(412 kg)Vf² - (1/2)(412 kg)(13.0 m/s)²
First, let's calculate the initial kinetic energy:
(1/2)(412 kg)(13.0 m/s)² = 0.5 * 412 kg * 169 m²/s² = 34834 J
Now, plug this back into the equation:
9899.25 J = 206 kg * Vf² - 34834 J
Add 34834 J to both sides:
44733.25 J = 206 kg * Vf²
Divide by 206 kg:
Vf² = 44733.25 J / 206 kg ≈ 217.15 m²/s²
Take the square root to find Vf:
Vf = √(217.15 m²/s²) ≈ 14.74 m/s
Therefore, the final velocity of the truck is approximately 14.74 m/s. We have successfully navigated through the physics principles and mathematical steps to arrive at our answer!
Final Velocity: 14.74 m/s
Alright, guys, we've reached the end of our physics journey! After a series of calculations, we've determined that the final velocity of the truck is approximately 14.74 m/s. This means that after the engine applied a 1005 N force over a distance of 9.85 meters, the truck sped up from its initial velocity of 13.0 m/s to about 14.74 m/s. We used a combination of Newton's Second Law and the Work-Energy Theorem to solve this problem, illustrating the powerful connections between force, work, and energy in physics. Remember, physics isn't just about formulas and equations; it's about understanding how the world around us works. By breaking down complex problems into smaller, manageable steps, we can tackle anything from calculating a truck's final velocity to understanding more complex phenomena. So, keep exploring, keep questioning, and keep applying those physics principles! You've got this! This problem not only gives us a numerical answer but also a deeper understanding of how forces and energy interact to change an object's motion. Now, let's recap the key steps and takeaways from this problem.
Key Takeaways
This problem provided a fantastic opportunity to apply fundamental physics principles to a real-world scenario. Here are some key takeaways:
- Newton's Second Law (F = ma) is crucial for understanding the relationship between force, mass, and acceleration. It allowed us to determine how the force applied by the truck's engine resulted in acceleration.
- The Work-Energy Theorem (W = ΔKE) connects the work done on an object to its change in kinetic energy. This theorem provided a direct link between the force applied over a distance and the change in the truck's velocity.
- Problem-solving in physics often involves breaking down complex scenarios into smaller, manageable steps. By identifying the knowns and unknowns, selecting the appropriate equations, and working through the math systematically, we can solve a wide range of problems.
- Units are essential in physics. Keeping track of units ensures that our calculations are consistent and our answers are meaningful. For instance, we made sure to use Newtons (N) for force, kilograms (kg) for mass, meters (m) for distance, and meters per second (m/s) for velocity.
- Real-world applications of physics concepts make learning more engaging and relevant. Understanding how to calculate the final velocity of a truck can help us appreciate the physics behind everyday experiences like driving a car or riding a bike.
By mastering these key takeaways, you'll be well-equipped to tackle similar physics problems and deepen your understanding of the world around you. Physics is all about understanding the fundamental laws that govern our universe, and every problem we solve brings us one step closer to that understanding. So, keep practicing, keep learning, and most importantly, keep having fun with physics!