Solve Relative Motion Problem Find The Speed Of A Man Meeting Taxis

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Let's delve into a classic problem of relative motion. Imagine a scenario at a taxi stand where two cabs depart at a speed of 74 km/hr, with a time interval of 28 minutes between their departures. Both cabs are traveling in the same direction along a straight route. Now, consider a man traveling in the opposite direction towards the taxi stand. This man encounters the cabs at regular intervals of 10 minutes. The challenge is to determine the speed at which the man is traveling.

This problem beautifully illustrates the concept of relative velocity and how it influences the perception of motion between different observers. We will explore the key principles governing relative motion and apply them to dissect this taxi encounter puzzle. By carefully analyzing the given information, we can unravel the unknowns and calculate the man's speed with precision.

To successfully tackle this problem, we must first grasp the core principles of relative velocity. Relative velocity is the velocity of an object as observed from a particular frame of reference. In simpler terms, it's how fast an object appears to be moving to someone who may also be in motion. When objects move in the same direction, their relative velocity is the difference between their individual velocities. Conversely, when objects move in opposite directions, their relative velocity is the sum of their individual velocities.

In our taxi stand scenario, we have three key players: the first cab, the second cab, and the man. The cabs are moving in the same direction, while the man is traveling in the opposite direction. This means that the relative velocity between the man and the cabs will be higher than the actual speed of either the man or the cabs. The time intervals at which the man encounters the cabs provide crucial clues about these relative velocities.

By carefully considering the relative motion between the man and the cabs, we can establish a relationship between their speeds and the time intervals. This will pave the way for setting up equations and ultimately solving for the man's speed. Let's proceed to break down the problem step-by-step and apply the principles of relative velocity to find the solution.

To solve this intriguing problem, we need to translate the given information into mathematical equations. Let's start by defining our variables:

  • Let Vm represent the speed of the man (in km/hr), which is what we aim to find.
  • The speed of the cabs is given as 74 km/hr.
  • The time interval between the cabs' departures is 28 minutes, which we'll need to convert to hours (28/60 hours).
  • The time interval between the man's encounters with the cabs is 10 minutes, which we'll also convert to hours (10/60 hours).

Now, let's consider the distance covered by the first cab in 28 minutes. Since distance is the product of speed and time, the distance between the two cabs when the second cab starts is (74 km/hr) * (28/60 hr). This distance remains constant throughout the problem.

The man meets the cabs at intervals of 10 minutes. In this 10-minute interval, the man covers a certain distance, and the first cab also covers a distance. The sum of these distances must be equal to the initial distance between the cabs. This is where the concept of relative velocity comes into play. The relative speed between the man and the cabs is the sum of their individual speeds (Vm + 74 km/hr).

Thus, we can set up the following equation:

(Relative speed) * (Time interval) = Initial distance between cabs

(Vm + 74) * (10/60) = 74 * (28/60)

This equation forms the cornerstone of our solution. By solving for Vm, we can determine the speed of the man.

With the equation established, the next step is to solve for Vm, the speed of the man. Let's revisit the equation:

(Vm + 74) * (10/60) = 74 * (28/60)

First, we can simplify the equation by canceling out the common factor of (1/60) on both sides:

(Vm + 74) * 10 = 74 * 28

Now, divide both sides by 10:

Vm + 74 = (74 * 28) / 10

Calculate the right side of the equation:

Vm + 74 = 207.2

Finally, subtract 74 from both sides to isolate Vm:

Vm = 207.2 - 74

Vm = 133.2 km/hr

Therefore, the speed of the man is 133.2 km/hr. This result highlights the significant impact of relative motion. The man's speed, as observed from the ground, is considerably high due to his movement in the opposite direction to the cabs.

In conclusion, we have successfully determined the speed of the man in this intriguing problem of relative motion. By meticulously analyzing the scenario, applying the principles of relative velocity, and setting up the appropriate equations, we arrived at the solution: the man is traveling at a speed of 133.2 km/hr.

This problem serves as a powerful illustration of how relative motion influences our perception of speed and time. It underscores the importance of considering the frame of reference when analyzing motion. The concept of relative velocity finds applications in various real-world scenarios, from air traffic control to navigation, making it a fundamental concept in physics and engineering.

Moreover, this exercise highlights the problem-solving process itself. By breaking down a complex problem into smaller, manageable steps, defining variables, establishing relationships, and applying mathematical tools, we can effectively tackle challenges and arrive at solutions. The journey of solving this problem has not only revealed the man's speed but also reinforced our understanding of relative motion and the power of analytical thinking.

Relative Motion Problem Solving Find the Speed of a Man Meeting Cabs

How to find the speed of the man in the taxi problem given the speed of cabs, time interval between cabs, and time interval between encounters?