Calculating Time For 100 Meters In Uniformly Accelerated Motion A Physics Problem Solution

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How long does it take for an athlete, starting from rest and undergoing uniformly accelerated motion (MRUV), to cover the first 100 meters if they cover 9 meters in 3 seconds?

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When delving into the fascinating world of physics, understanding the principles of motion is paramount. One such principle is uniformly accelerated motion (MRUV), which describes the movement of an object with a constant acceleration. Let's explore a classic problem involving an athlete accelerating from rest and calculate the time it takes to cover a specific distance.

Problem Statement

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An athlete starts from rest and undergoes uniformly accelerated motion (MRUV), covering 9 meters in 3 seconds. The question is: how long will it take for the athlete to cover the first 100 meters?

Understanding Uniformly Accelerated Motion (MRUV)

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Before we dive into the solution, let's clarify the fundamental concepts of MRUV. In MRUV, an object's velocity changes at a constant rate. This constant rate of change in velocity is known as acceleration. The key equations that govern MRUV are:

  1. Displacement (d): d = v₀t + (1/2)at²
  2. Final Velocity (v): v = v₀ + at
  3. Velocity-Displacement Relation: v² = v₀² + 2ad

Where:

  • d represents the displacement or distance traveled.
  • v₀ denotes the initial velocity.
  • v signifies the final velocity.
  • a stands for acceleration.
  • t represents the time elapsed.

These equations provide the framework for analyzing and solving problems related to uniformly accelerated motion. It's crucial to understand these equations to tackle the problem at hand effectively. Mastering these formulas will allow you to predict and calculate various parameters of motion, such as displacement, velocity, and time, under constant acceleration.

Solving the Problem Step-by-Step

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To solve the problem of the athlete covering 100 meters, we'll follow a step-by-step approach, utilizing the MRUV equations we discussed earlier. This structured method will ensure clarity and accuracy in our solution.

Step 1: Determine the Acceleration

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In this initial step, our primary goal is to calculate the athlete's acceleration. We are given that the athlete starts from rest (v₀ = 0 m/s), covers a distance of 9 meters (d = 9 m), and takes 3 seconds (t = 3 s) to do so. We can use the displacement equation to find the acceleration:

d = v₀t + (1/2)at²

Substituting the given values:

9 m = (0 m/s)(3 s) + (1/2)a(3 s)²

Simplifying the equation:

9 m = (1/2)a(9 s²)

To isolate 'a', we multiply both sides by 2:

18 m = a(9 s²)

Now, we divide both sides by 9 s² to solve for 'a':

a = 18 m / (9 s²) = 2 m/s²

Therefore, the athlete's acceleration is 2 m/s². This means that the athlete's velocity increases by 2 meters per second every second. Understanding the concept of acceleration is crucial in physics, as it describes how the velocity of an object changes over time. In this case, the constant acceleration allows us to predict the athlete's motion accurately using the MRUV equations. This step sets the foundation for calculating the time it takes to cover 100 meters.

Step 2: Calculate the Time to Cover 100 Meters

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Now that we have determined the acceleration (a = 2 m/s²), we can proceed to calculate the time it will take for the athlete to cover 100 meters. We will again use the displacement equation, but this time, we are solving for 't' when d = 100 m. The initial velocity remains the same (v₀ = 0 m/s) since the athlete starts from rest.

The displacement equation is:

d = v₀t + (1/2)at²

Substituting the known values:

100 m = (0 m/s)t + (1/2)(2 m/s²)t²

Simplifying the equation:

100 m = (1 m/s²)t²

To solve for t², we divide both sides by 1 m/s²:

t² = 100 m / (1 m/s²) = 100 s²

Now, we take the square root of both sides to find 't':

t = √(100 s²) = 10 s

Therefore, it will take the athlete 10 seconds to cover the first 100 meters. This calculation highlights the power of the MRUV equations in predicting motion. By understanding the relationships between displacement, velocity, acceleration, and time, we can solve a wide range of physics problems. The result of 10 seconds is a direct consequence of the athlete's constant acceleration and the distance they need to cover.

Alternative Approach: Using Velocity-Displacement Relation

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While we successfully calculated the time using the displacement equation, there's an alternative approach using the velocity-displacement relation. This method offers a different perspective and can be useful in various scenarios. Let's explore this alternative approach.

Step 1: Find the Final Velocity After 100 Meters

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Using the velocity-displacement relation, we can directly find the final velocity (v) after the athlete covers 100 meters. The velocity-displacement relation is:

v² = v₀² + 2ad

We know that the initial velocity (v₀) is 0 m/s, the acceleration (a) is 2 m/s², and the displacement (d) is 100 m. Substituting these values:

v² = (0 m/s)² + 2(2 m/s²)(100 m)

Simplifying the equation:

v² = 0 + 400 m²/s²

v² = 400 m²/s²

Now, we take the square root of both sides to find 'v':

v = √(400 m²/s²) = 20 m/s

So, the athlete's final velocity after covering 100 meters is 20 m/s. This step demonstrates the utility of the velocity-displacement relation in directly linking velocity, acceleration, and displacement without explicitly involving time. It's a valuable tool in solving MRUV problems when time is not directly given or sought in the initial step.

Step 2: Calculate the Time Using the Final Velocity

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Now that we know the final velocity (v = 20 m/s), we can use the final velocity equation to calculate the time (t) it takes to reach this velocity. The final velocity equation is:

v = v₀ + at

We know that the initial velocity (v₀) is 0 m/s and the acceleration (a) is 2 m/s². Substituting these values:

20 m/s = 0 m/s + (2 m/s²)t

Simplifying the equation:

20 m/s = (2 m/s²)t

To solve for 't', we divide both sides by 2 m/s²:

t = (20 m/s) / (2 m/s²) = 10 s

Again, we find that it takes the athlete 10 seconds to cover the first 100 meters. This alternative approach confirms our previous result and showcases the flexibility in solving MRUV problems. By using different equations and approaches, we can gain a deeper understanding of the underlying physics and verify our solutions.

Conclusion

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In conclusion, by applying the principles of uniformly accelerated motion and utilizing the appropriate equations, we determined that it will take the athlete 10 seconds to cover the first 100 meters. We solved this problem using two different methods: first, by directly using the displacement equation, and second, by employing the velocity-displacement relation to find the final velocity and then calculating the time. Both methods yielded the same result, highlighting the consistency and reliability of the MRUV equations.

Understanding MRUV is crucial for analyzing various real-world scenarios involving motion under constant acceleration. From the motion of vehicles to the trajectory of projectiles, the principles of MRUV provide a powerful framework for understanding and predicting motion. By mastering these concepts, you'll be well-equipped to tackle a wide range of physics problems and gain a deeper appreciation for the world around you. This problem serves as an excellent example of how theoretical physics can be applied to practical situations, allowing us to make quantitative predictions about the motion of objects.

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