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© 1995 University of Massachusetts, Amherst
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9730438
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a2l, assessment, audience response, classroom vote, clicker question, concept test, eliciting understanding, peer instruction, reponse system, student learning
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Metadata instance created July 5, 2006 by Ian Beatty
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June 21, 2012 by Bruce Mason
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when Cataloged:
June 30, 2007
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item 152

Author: SO HING NAM
Posted: September 16, 2009 at 10:10AM
Source: The PSRC collection

I want to point out that ball B will act as a horizontal projectile after passing the dip. If so, the situation is a bit complicated and thus we can't make any conclusion.

» reply

Re: item 152

Author: Bruce, ComPADRE Dir
Posted: Sep 16, 2009 at 12:10PM

There is an approximation assumed here that the ball stays on the track, or if it leaves the track it is for a very short period of time. If the ball is rolling fairly slowly this is quite reasonable. Try it. We're also approximating that this process conserves energy.

However, you are not correct that if the ball goes into the air the predictions are no longer valid. As soon as the ball lands back on the track, there will be an acceleration in the x-direction meaning the ball on track B will have a larger velocity in the x-direction than the ball on track A. This velocity (x-velocity of ball B) will be larger than the velocity of ball A for the entire time until the tracks are back on the same level. This is a very general principle that really requires only energy conservation, or something close to it.

In fact, this is related to a classic problem that was first studied centuries ago - given initial and final positions and velocities of an object in 2D, what is the path of least time? It is not the horizontal, constant velocity path.

A video of a real experiment like this (although the ball doesn't leave the track) is at:

http://www.physics.umd.edu/lecdem/services/demos/demosc2/c2-11.mpg

> On Sep 16, 2009, SO HING NAM posted:
>
> I want to point
> out that ball B will act as a horizontal projectile
> after passing the dip. If so, the situation is a bit
> complicated and thus we can't make any conclusion.

» reply

Re: item 152

Author: SO HING NAM
Posted: Sep 21, 2009 at 1:45PM

First of all, Thanks for your reply. You have tried your best to explain. But your arguement about the velocity in x-direction for the whole process of ball B after projected horizontally and make collision with the slope,  Do you think the collision is elastic? How many times will ball B bounce back from the slope. If it is uncertain, the situation will be too complicated and no conclusion can be made.
I fully understand the famous old problem that you have mentioned, However the situation is not the same. It start at rest and prevent any collision on the track. Please reply.  





> On Sep 16, 2009, SO HING NAM posted:
>
> I want to point
> out that ball B will act as a horizontal projectile
> after passing the dip. If so, the situation is a bit
> complicated and thus we can't make any conclusion.

» reply

Item 152

Author: SO HING NAM
Posted: September 13, 2009 at 9:10PM
Source: The PSRC collection

Obviously, The x component of the initial velocity of the ball B is smaller than ball A. Although it has an acceleration in x direction, it needs time to speed up. Hence, we can't conclude that the average velocity in x direction is higher for ball B.
Do you agree?

» reply

Re: Item 152

Author: Bruce, ComPADRE Dir
Posted: Sep 13, 2009 at 11:08PM

Why would the initial velocity of ball B in the x-direction be smaller? Both A and B start at the same height, so on the horizontal part of the track, they will both have the same velocity (in the x-direction). Once the ball on track B hits the dip, it will accelerate in both x and y.

Ball B's x velocity will be greater than a ball on track A, and can never be smaller simply by energy conservation.

> On Sep 13, 2009, SO HING NAM posted:
>
> Obviously, The
> x component of the initial velocity of the ball B
> is smaller than ball A. Although it has an acceleration
> in x direction, it needs time to speed up. Hence,
> we can't conclude that the average velocity in x direction
> is higher for ball B.
> Do you agree?

» reply

Re: Re: Item 152

Author: SO HING NAM
Posted: Sep 14, 2009 at 12:12AM

I want to make sure that by energy conservation, when ball B slide down the slope, its initial speed is the same as ball A. However, its direction is along the slope. So the component of initial velocity in x dirction will be smaller than ball A. Is this right? Please reply. Thanks.










> On Sep 13, 2009, Bruce, ComPADRE Dir posted:
>
> Why
> would the initial velocity of ball B in the x-direction
> be smaller? Both A and B start at the same height,
> so on the horizontal part of the track, they will
> both have the same velocity (in the x-direction).
> Once the ball on track B hits the dip, it will accelerate
> in both x and y.
>
> Ball B's x velocity will be greater
> than a ball on track A, and can never be smaller simply
> by energy conservation.
>
> > On Sep 13, 2009, SO HING
> NAM posted:
> >
> > Obviously, The
> > x component of the
> initial velocity of the ball B
> > is smaller than ball
> A. Although it has an acceleration
> > in x direction,
> it needs time to speed up. Hence,
> > we can't conclude
> that the average velocity in x direction
> > is higher
> for ball B.
> > Do you agree?

» reply

Re: Re: Re: Item 152

Author: SO HING NAM
Posted: Sep 14, 2009 at 12:25AM

In addition, is track B smooth enough to allow Ball B slide on the track? That means there isn't any flying motion for ball B after hitting the dip.




> On Sep 14, 2009, SO HING NAM posted:
>
> I want to make
> sure that by energy conservation, when ball B slide
> down the slope, its initial speed is the same as ball
> A. However, its direction is along the slope. So the
> component of initial velocity in x dirction will be
> smaller than ball A. Is this right? Please reply. Thanks
>
>
>
>
>
>
>
>
>
>
>
> > On Sep 13, 2009, Bruce, ComPADRE Dir posted
>
> >
> > Why
> > would the initial velocity of ball B in
> the x-direction
> > be smaller? Both A and B start at
> the same height,
> > so on the horizontal part of the
> track, they will
> > both have the same velocity (in
> the x-direction).
> > Once the ball on track B hits
> the dip, it will accelerate
> > in both x and y.
> >
>
> > Ball B's x velocity will be greater
> > than a ball
> on track A, and can never be smaller simply
> > by energy
> conservation.
> >
> > > On Sep 13, 2009, SO HING
> > NAM
> posted:
> > >
> > > Obviously, The
> > > x component of th
>
> > initial velocity of the ball B
> > > is smaller than bal
>
> > A. Although it has an acceleration
> > > in x direction
>
> > it needs time to speed up. Hence,
> > > we can't conclud
>
> > that the average velocity in x direction
> > > is highe
>
> > for ball B.
> > > Do you agree?

» reply

Re: Re: Re: Re: Item 152

Author: Bruce, ComPADRE Dir
Posted: Sep 14, 2009 at 10:40AM

> On Sep 14, 2009, SO HING NAM posted:
>
> In addition,
> is track B smooth enough to allow Ball B slide on
> the track? That means there isn't any flying motion
> for ball B after hitting the dip.
>

Yes, the approximation here is that there are no collisions or other events that will result in loss of mechanical (kinetic + potential) energy.

» reply

Re: Re: Re: Item 152

Author: Bruce, ComPADRE Dir
Posted: Sep 14, 2009 at 10:48AM

> On Sep 14, 2009, SO HING NAM posted:
>
> I want to make
> sure that by energy conservation, when ball B slide
> down the slope, its initial speed is the same as ball
> A. However, its direction is along the slope. So the
> component of initial velocity in x dirction will be
> smaller than ball A. Is this right? Please reply. Thanks

The ball rolls down the initial slope and reaches the horizontal part of both tracks A and B. The velocity, call it v0, is the same for both tracks.

On track A, the velocity of the ball will stay at v0 for the entire trip.

On track B, right when the ball reaches the second slope, its x velocity is v0 and its y velocity is 0. As it starts to roll down the slope, there is acceleration in both the x and y directions, but initial velocity is still v0 in the x direction and 0 in the y direction.

The velocity is a vector, so the change in velocity in each direction is equal to the acceleration in each direction. The velocity can not suddenly change from horizontal to down the slope, with the same magnitude or speed, as that would require infinite accelerations and forces.

» reply

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