Incline question

[unleash500]

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Why is it that when two balls roll down two inclines with different angles, that they have the same final velocity?

I realize that the larger the angle the larger the magnitude of acceleration but why is it that v is the same at the end?

I also know that the time it takes for the ball to roll down is shorter when the angle is larger.

Furthermore, the distance x is also longer for those with a smaller angle.

Putting this all together, with changing angles, we change the acceleration, the distance , and the time it takes to fall.

Yet, why is it that velocity is the one that is constant here? (I am looking for a physical explanation, as I can see how the math works out)

 

[echolalia16]

The ball that rolls down the steeper incline will accelerate at a faster rate, but will overall spend less time accelerating on the incline. The ball that rolls with the smaller incline will accelerate more slowly, but have more time to accumulate a greater velocity. Does that help at all? I feel like you answered your own question.

 

[PiBond]

Why is it that when two balls roll down two inclines with different angles, that they have the same final velocity?

I realize that the larger the angle the larger the magnitude of acceleration but why is it that v is the same at the end?

I also know that the time it takes for the ball to roll down is shorter when the angle is larger.

Furthermore, the distance x is also longer for those with a smaller angle.

Putting this all together, with changing angles, we change the acceleration, the distance , and the time it takes to fall.

Yet, why is it that velocity is the one that is constant here? (I am looking for a physical explanation, as I can see how the math works out)

I think of it in terms of conservation of energy. We know that Ug = 1/2mv^2 and that mathematically the masses don't matter. Therefore, assuming that both balls start as the same height they will have all of their gravitational potential energy converted into kinetic energy at the bottom of the incline. Thus they have the same velocity

Also, since one is traveling a greater x-distance it has a longer time to speed up going down the incline versus the ball going down the steeper incline. Therefore, even though the ball on the steeper incline will have a shorter time down the slope the other has more time to catch-up (velocity wise).

 

[unleash500]

thank you pi bond you answered my question!
I always forget to think in terms of energy.
Thanks for the replies.