What Is the Average Vertical Velocity Halfway Through a Circular Path?

[ortegavs]

Homework Statement

A particle moving from position 1 to position 2 moves along path C ( which is half a circle ). It travels at a constant speed of 5 pi m/s. At exactly half way ( 1/4 of a circle ) through the trip its average vertical velocity is? Radius is 5 meters.

Homework Equations

speed=distance/time
circumference=2 pi r
V avg= Vi + Vf/2
Vf=x/t

Unknowns t, Vavg, Vf

The Attempt at a Solution

I assumed that Vi is zero which simplifies my kinematic equation to V avg=Vf/2. I then substituted x/t for Vf to get Vavg=x/2t. I solved for t using the speed equation. t= distance/speed and then distance=πr/2=π5/2 and speed is 5π. The 5π cancels and the equation simplified to t=1/2s I then plugged in my value for t into the kinematic equation which gives x/2t=x/2/2 since x=5 then 5/2/2 and this gives you 5 m/s. The answer given is 10 m/s but this answer is the final velocity not the average. Am I right about this or does the book have right? The only thing that bothers me is that I set Vi to zero but the question says constant speed meaning that Vi and Vf should be the same that is 5π m/s at the beginning and at the end. Still the answer would not be 10 m/s.

 

[PhanthomJay]

The book answer is correct, assuming points A and B lie on a horizontal axis (you didn't show a picture). You must first look at the definition of average vertical velocity. Average vertical velocity between 2 points is defined as total vertical displacement between those points divided by total elapsed time during that displacement. What is the total vertical displacement after it has traveled the 1/4 circle arc? What time has elapsed?

 

[ortegavs]

Jay

Points A and B do lie on a horiz. axis. The definition of average V is vi +vf/2. I don't understand why I can't use this definition? And if you use to points A and B you have to divide by two I don't understand. So with the your definition V avg= xf-xi/t which is 5/1/2 and this equal to 10 m/s which is the book answer. How do you know which definition to use?

 

[ortegavs]

Ok I think I got it. Since the object is moving in a circle then it is constantly changing its direction thus its acceleration is not constant as I had assumed. Given this I cannot use kinematic equation V avg = Vf+Vi/2 since this equation requires constant a. Thus I have to us the other definition provide by Jay V avg= displacement/time

Finally it makes sense

 

[PhanthomJay]

Ok I think I got it. Since the object is moving in a circle then it is constantly changing its direction thus its acceleration is not constant as I had assumed. Given this I cannot use kinematic equation V avg = Vf+Vi/2 since this equation requires constant a. Thus I have to us the other definition provide by Jay V avg= displacement/time

Finally it makes sense

Yes, that is correct, nice work. Both definitions of average velocity are the same only when acceleration is constant. Note, however , that the result of 10 m/s is the average vertical velocity (in the y direction), which is vertical displacement/time, which is what the problem asked, and which is therefore correct. I want to point out, however, that the average velocity would be different. The total displacement, d, would be (R)(sq rt 2) = 5(1.414) = 7.07 m, and the average velocity would be d/t = (7.07)/(1/2) = 14.14 m/s at an angle of 45 degrees with the horizontal. The reader should fully understand this before proceeding any further