How Is Average Velocity Calculated in Multi-Stage Vector Displacements?

[Sullivan]

You go for a short walk traveling in three stages. The first displacement is 58.5 m 20.0 degrees east of north. The second displacement is 78.0 m 40.0 degrees south of east. Finally you go 99.0 m 17.0 degrees north of west. The answer I got was 36.9 m, 66.2 degrees north of west,which is correct.

But then they ask: If this trip took 125 seconds, what was the average velocity? They give the answer: 0.295 m/s, 66.2 degrees north of west.

What I got for that answer was 1.884 m/s. I figured you could just use the V avg = total displacement over time equation. So 58.5 + 78 + 40 / 125 = 1.844.

What am I doing wrong here?

Thanks for the help!

 

[Gauss]

Hey Sullivan.

Velocity is a vector function, so direction is important. Displacement refers to the distance between an initial and final point; in this case, your initial point is where you start and your final point is where you end. You found that the distance between these two points is 36.9 m. So, in the period of 125 seconds, you moved to a position 36.9 m from your initial position.

Velocity is found by dividing the displacement by time. so you have to divide 36.9/125 = .2952 m/s.

Anything unclear?

 

[Sullivan]

Right! Thank you very much. I was using total distance rather than displacement wasn't I!

 

[Gauss]

Yep! You were calculating the speed as opposed to the velocity, these two are often confused as are distance and displacement.

 

[HalfLostAndSearching]

Your calculation for the average velocity is incorrect. The equation Vavg = total displacement over time is only applicable for one-dimensional motion. In this case, the motion is two-dimensional as the displacements are given in both east-west and north-south directions. Therefore, you need to use vector addition to find the total displacement, and then divide by the total time to get the average velocity.

To find the total displacement, you need to add the individual displacements in both magnitude and direction. This can be done using the Pythagorean theorem and trigonometric functions. The correct calculation for the total displacement is:

√(58.5^2 + 78^2 + 99^2 + 2(58.5)(78)cos(40) + 2(58.5)(99)cos(107) + 2(78)(99)cos(57)) = 117.6 m

To find the total time, you can simply add the times for each stage, which is 125 seconds.

Therefore, the correct calculation for the average velocity is:

Vavg = 117.6 m / 125 sec = 0.941 m/s

This is the magnitude of the average velocity. To find the direction, you can use trigonometric functions again to find the angle between the total displacement vector and the north-west direction. This angle is 66.2 degrees, as calculated in the given answer.

In conclusion, your mistake was using the Vavg = total displacement over time equation, which is only applicable for one-dimensional motion. For two-dimensional motion, you need to use vector addition to find the total displacement and then divide by the total time to find the average velocity.