The Problem Discusses Two Mobiles, (1) And (2), Starting Simultaneously From An Indicated Position With Accelerations Of 5 M/s² And 3 M/s², Respectively. The Mobiles Have MRUV Motion. The Question Asks To Determine The Time It Takes For Mobile (1) To Reach Mobile (2), Considering The Mobiles Have An Initial Separation Of 25 M.

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Introduction

In physics, understanding the motion of objects is fundamental. When dealing with objects moving with constant accelerations, we delve into the realm of uniformly accelerated motion (MRUV). This article addresses a classic problem in kinematics: determining the time it takes for one object to overtake another when both are moving with different constant accelerations. Let's analyze the scenario and derive the solution step-by-step.

Problem Statement

Consider two objects, mobile (1) and mobile (2), starting simultaneously from the same initial position. Mobile (1) has an acceleration a₁ = 5 m/s², while mobile (2) has an acceleration a₂ = 3 m/s². Our goal is to determine the time it takes for mobile (1) to overtake mobile (2).

Understanding Uniformly Accelerated Motion (MRUV)

Before we dive into the solution, it's crucial to understand the key concepts of MRUV. In MRUV, the velocity of an object changes at a constant rate. This constant rate of change is what we call acceleration. The fundamental equations governing MRUV are:

  1. v = u + at (where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time).
  2. s = ut + (1/2)at² (where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time).
  3. v² = u² + 2as (where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement).

These equations provide the foundation for analyzing and solving problems related to motion with constant acceleration. In our problem, we will primarily use the second equation to relate displacement, acceleration, and time.

Setting up the Equations

Let's denote the time it takes for mobile (1) to overtake mobile (2) as t. At this time, both mobiles will have covered the same distance. Let's denote this distance as s. We can now write equations for the displacement of each mobile using the second equation of MRUV.

For mobile (1):

Since mobile (1) starts from rest, its initial velocity u₁ = 0 m/s. The displacement s₁ after time t is given by:

s₁ = u₁t + (1/2)a₁t² = 0t + (1/2)(5)t² = 2.5t²

For mobile (2):

Similarly, mobile (2) also starts from rest, so its initial velocity u₂ = 0 m/s. The displacement s₂ after time t is given by:

s₂ = u₂t + (1/2)a₂t² = 0t + (1/2)(3)t² = 1.5t²

Solving for Time

When mobile (1) overtakes mobile (2), they will have traveled the same distance. Therefore, we can equate the displacements s₁ and s₂:

s₁ = s₂

*2.5t² = 1.5t²

Now, we can solve for t:

2. 5t² - 1.5t² = 0

t² = 25

t = √25 = 5 s

t = √(s²)

t = 5 s

However, this is an incomplete solution. We missed a crucial aspect: mobile (2) has an initial 25m head start. Let's correct our approach.

Corrected Approach: Accounting for the Initial Separation

The problem states that the mobiles start from the indicated position, but there's an initial separation of 25 meters between them. This means mobile (2) effectively has a head start. We need to incorporate this into our equations.

The equation for mobile (1) remains the same:

s₁ = 2.5t²

For mobile (2), we need to add the initial separation to its displacement:

s₂ = 25 + 1.5t²

Now, equating the displacements:

2. 5t² = 25 + 1.5t²

Solving for t:

t² = 25

t = √25 = 5

t² = 25

t = √(25)

t = 5 s

Detailed Solution with Equations

To solve this problem accurately, let's break down the steps and use the equations of motion for uniformly accelerated motion (MRUV).

Step 1: Define Variables

  • a₁ = Acceleration of mobile (1) = 5 m/s²
  • a₂ = Acceleration of mobile (2) = 3 m/s²
  • d₀ = Initial distance between the mobiles = 25 m
  • t = Time taken for mobile (1) to reach mobile (2) (what we need to find)
  • x₁ = Distance traveled by mobile (1) when it reaches mobile (2)
  • x₂ = Distance traveled by mobile (2) when it is reached by mobile (1)

Step 2: Equations of Motion

The equation for the distance traveled (x) in MRUV is given by:

x = v₀t + (1/2)at²

Where:

  • v₀ is the initial velocity
  • t is the time
  • a is the acceleration

Since both mobiles start from rest, their initial velocities (v₀) are 0.

Step 3: Distance Traveled by Mobile (1)

The distance traveled by mobile (1) is:

x₁ = (1/2)a₁t² = (1/2) * 5 * t² = 2.5t²

Step 4: Distance Traveled by Mobile (2)

The distance traveled by mobile (2) is:

x₂ = (1/2)a₂t² = (1/2) * 3 * t² = 1.5t²

But mobile (2) also has an initial head start of 25 m, so we add this to the distance:

x₂ = 1.5t² + 25

Step 5: Equating the Distances

When mobile (1) reaches mobile (2), the distances they've traveled (including the initial separation) will be equal:

x₁ = x₂

2. 5t² = 1.5t² + 25

Step 6: Solve for t

Subtract 1.5t² from both sides:

2. 5t² - 1.5t² = 25

t² = 25

t = ±√25

Since time cannot be negative, we take the positive root:

t = 5 s

Final Answer

Therefore, mobile (1) will reach mobile (2) after 5 seconds. The correct answer is A) 5 s.

Conclusion

This problem demonstrates a practical application of the principles of uniformly accelerated motion. By carefully setting up the equations of motion and considering the initial conditions, we can accurately predict the time it takes for one object to overtake another. Understanding these concepts is crucial for solving a wide range of physics problems related to motion.

In summary, solving problems involving uniformly accelerated motion requires a solid understanding of the equations of motion and careful consideration of initial conditions. This example highlights the importance of accounting for initial separations and using the appropriate equations to determine the time of overtaking. Mastering these concepts is essential for further studies in physics and related fields. To solidify your grasp, practice similar problems with varying accelerations and initial conditions. This will strengthen your problem-solving skills and enhance your understanding of uniformly accelerated motion. Furthermore, exploring real-world applications of these concepts, such as analyzing the motion of vehicles or projectiles, can provide valuable insights and practical context. Understanding the relationship between acceleration, velocity, and displacement is key to mastering kinematics. Keep practicing, and you'll become proficient in solving complex motion problems.