Average velocity from multiple displacement/velocities

[BogMonkey]

Heres the problem:
"A car travels at a constant speed of 60kph for 30km, 40kph for another 30km and 50kph for the final 30km. What is the average speed of the car."

I know that in this case I can just add the velocities and divide by 3 since the displacements are equal and get 50kph as the average speed but when I tried to do it the long way I didn't get the same answer. First I calculated the total time elapsed like this d1/v1 + d2/v2 + d3/v3 and got t = 1.85h. Then to get the average speed I divided the total displacement by time (90/1.85) and got 48.65kph.

Theres a pattern there I've tried this with various problems and I always a little less than the right answer. Whats going on here?

 

[Doc Al]

I know that in this case I can just add the velocities and divide by 3 since the displacements are equal and get 50kph as the average speed

Although it might seem right, it's not. You really need to find total displacement over time.

Now if the time intervals were equal, then it would work.

 

[astrokreng]

I would first commend you for taking the time to explore and understand the concept of average velocity. It is important to not just rely on formulas, but to also understand the underlying principles behind them.

To address the discrepancy in your calculations, it is important to understand the difference between average speed and average velocity. Average speed is the total distance traveled divided by the total time elapsed, while average velocity is the total displacement divided by the total time elapsed.

In this problem, the car's average speed is indeed 50kph, as you correctly calculated by adding the velocities and dividing by 3. However, the average velocity is slightly different because the car is not traveling in a straight line. It starts and ends at the same point, but during the middle portion, it is traveling in a different direction. This means that the displacement is not simply the sum of the distances traveled, but it also takes into account the direction of movement.

To calculate the average velocity, we need to consider the displacement vectors for each segment of the journey. In this case, the first and third segments have a displacement of 30km in the same direction, while the second segment has a displacement of 30km in the opposite direction. This results in a total displacement of 30km in the original direction, which is why the average velocity is slightly less than the average speed.

In summary, the discrepancy in your calculations is due to the difference between average speed and average velocity, and the fact that the car is not traveling in a straight line. It is important to consider both distance and direction when calculating average velocity.