Solving For Velocity V An Average Speed Problem In Physics
In the fascinating world of physics, understanding motion is paramount. One of the fundamental concepts in kinematics, the branch of physics dealing with motion, is average speed. Average speed, as the name suggests, represents the overall speed of an object over a given distance, considering variations in speed during the journey. This article delves into a classic problem involving average speed, challenging us to calculate an unknown velocity. We will dissect the problem, apply relevant formulas, and arrive at a solution, enhancing our grasp of motion-related concepts.
Problem Statement: The 200-Meter Journey
The scenario presents a car traversing a total distance of 200 meters. This journey is divided into two equal halves: the first 100 meters and the second 100 meters. The car covers the initial half at a constant speed of 40 kilometers per hour (km/h). The challenge lies in determining the speed (represented by the variable V) at which the car travels the second half of the distance, given that the overall average speed for the entire 200-meter journey is 48 km/h. This problem exemplifies how average speed is influenced by varying speeds over different segments of a journey. It necessitates a careful consideration of time, distance, and speed relationships.
Understanding Average Speed: A Key Concept
Before diving into the calculations, it's essential to solidify our understanding of average speed. Average speed is not simply the arithmetic mean (average) of the speeds during different parts of the journey. Instead, it's defined as the total distance traveled divided by the total time taken. This distinction is crucial because it accounts for the time spent at each speed, giving a more accurate representation of the overall motion. For instance, if a car travels at a high speed for a short duration and a low speed for a longer duration, the average speed will be closer to the lower speed due to the greater time spent at that speed. The formula for average speed is expressed as:
Average Speed = Total Distance / Total Time
This formula serves as the foundation for solving our problem. We need to determine the total time taken for the 200-meter journey, considering the different speeds and distances involved.
Deconstructing the Problem: Breaking Down the Journey
To solve for V, we need to break down the problem into manageable steps. The key is to express the total time taken for the journey in terms of the given speeds and distances. Since the journey is divided into two halves, we can calculate the time taken for each half separately and then add them to find the total time. Let's denote the time taken for the first half (100 meters at 40 km/h) as t1 and the time taken for the second half (100 meters at speed V) as t2. Our goal is to find an expression for t1 and t2 in terms of the known quantities and the unknown V.
Calculating Time for the First Half (t1)
For the first half of the journey, we know the distance (100 meters) and the speed (40 km/h). However, the units are inconsistent (meters and kilometers per hour). To maintain consistency, we need to convert the speed to meters per second (m/s). The conversion factor is 1 km/h = 1000/3600 m/s, which simplifies to 5/18 m/s. Therefore, 40 km/h is equal to 40 * (5/18) = 100/9 m/s. Now we can use the formula:
Time = Distance / Speed
to calculate t1:
t1 = 100 meters / (100/9 m/s) = 9 seconds
So, the car takes 9 seconds to cover the first 100 meters.
Expressing Time for the Second Half (t2) in Terms of V
For the second half, the distance is also 100 meters, but the speed is V (in km/h). We need to convert V to m/s by multiplying by 5/18, giving us V * (5/18) m/s. Using the same formula as before:
t2 = 100 meters / (V * (5/18) m/s) = 360 / V seconds
Thus, the time taken for the second half is expressed as 360/V seconds.
Applying the Average Speed Formula: The Final Calculation
Now that we have expressions for t1 and t2, we can use the average speed formula to solve for V. The total distance is 200 meters, and the average speed is 48 km/h. Converting the average speed to m/s, we get 48 * (5/18) = 40/3 m/s. The total time is t1 + t2 = 9 + (360/V) seconds. Plugging these values into the average speed formula:
40/3 m/s = 200 meters / (9 + 360/V) seconds
Now we have an equation with one unknown, V. Let's solve for V step by step:
- Multiply both sides by (9 + 360/V): (40/3) * (9 + 360/V) = 200
- Distribute the (40/3): 120 + 4800/V = 200
- Subtract 120 from both sides: 4800/V = 80
- Multiply both sides by V: 4800 = 80V
- Divide both sides by 80: V = 60
Therefore, the value of V is 60 km/h. This means the car traveled the second half of the distance at a speed of 60 km/h.
Conclusion: Mastering Motion through Problem-Solving
This problem beautifully illustrates the application of the average speed concept in physics. By carefully breaking down the journey into segments, calculating the time taken for each segment, and applying the average speed formula, we successfully determined the unknown velocity V. The key takeaways from this exercise are the importance of unit consistency, the correct application of the average speed formula, and the power of breaking down complex problems into simpler steps. Understanding average speed is crucial for analyzing motion in various scenarios, from everyday experiences like driving a car to more complex situations in physics and engineering. Through practice and problem-solving, we can strengthen our understanding of motion and the world around us. Remember, practice makes perfect when it comes to mastering these fundamental physics concepts. By working through similar problems, you can solidify your understanding and build your problem-solving skills. Keep exploring the fascinating world of physics, and you'll continue to unravel the mysteries of the universe!