Calculating the Speed of a Car Passing Cyclists at Different Speeds

In this article, we will delve into a practical example of how to calculate the speed of a car passing cyclists moving at different speeds. This problem showcases the application of relative speed concepts in solving real-world scenarios, making it an excellent example for students and professionals alike in fields such as physics, engineering, and data science.

Problem Statement

A and B go cycling in the same direction with speeds of 6 km/hr and 12 km/hr, respectively. A car from behind passes them in 9 and 10 seconds, respectively. What is the speed of the car?

Step-by-Step Solution

To solve this problem, we will use the concept of relative speed. Relative speed is the difference between the speed of the car and the speed of the cyclists.

Step 1: Calculate the Distance Covered by the Car While Passing A and B

Let's denote the speed of the car as v km/hr.

Distance Covered by the Car While Passing A

Time: 9 seconds
Speed of A: 6 km/hr

The relative speed of the car with respect to A is v - 6 km/hr. Converting 9 seconds to hours:

Distance (v - 6) * (9 / 3600) km

Distance Covered by the Car While Passing B

Time: 10 seconds
Speed of B: 12 km/hr

The relative speed of the car with respect to B is v - 12 km/hr. Converting 10 seconds to hours:

Distance (v - 12) * (10 / 3600) km

Step 2: Set the Distances Equal to Each Other

Since the distance covered by the car while passing A and B must be the same, we set the two equations equal:

(v - 6) * (9 / 3600) (v - 12) * (10 / 3600)

Step 3: Simplify the Equation

We can cancel out the 1 / 3600 from both sides:

v - 6 * 9 v - 12 * 10

Expanding both sides:

9v - 54 10v - 120

Step 4: Solve for v

Rearranging the equation gives:

120 - 54 10v - 9v

66 v

Therefore, the speed of the car is 66 km/hr.

Alternative Method Using Conversion to m/s

A simpler way to solve the same problem involves converting the speeds from km/hr to m/s:

v1 6 km/hr 5/3 m/s

v2 12 km/hr 10/3 m/s

At t1 9 seconds, the distance covered by A (x1) is:

x1 v1 * t1 (5/3) * 9 15 meters

At t2 10 seconds, the distance covered by B (x2) is:

x2 v2 * t2 (10/3) * 10 100/3 meters

To find the speed of the car (v), we use the relative speed:

v (x2 - x1) / (t2 - t1) (100/3 - 15) / 1 100/3 - 45/3 55/3 m/s

Converting back to km/hr:

v (55/3) * (3600/1000) 66 km/hr

Conclusion

The speed of the car is 66 km/hr. This example demonstrates the application of relative speed concepts in real-life scenarios, which are crucial in various fields, including physics and engineering.