The Flintstones and Rubbles decide to try out the new inclined bowling alley, ``

Question

The Flintstones and Rubbles decide to try out the new inclined bowling alley, ``Bedslant Bowling''. Betty's ball and Wilma's ball have the same size, but Wilma's ball is hollow. Fred's ball and Barney's ball are scaled down versions of Betty's ball and Wilma's ball respectively. They all place their bowling balls on the same pitch incline and release them from rest at the same time. Answer choices are (greater than, less than or equal to)

The time it takes for Wilma's ball to hit the pins is .... that for Barney's ball to hit.

The time it takes for Barney's ball to hit the pins is .... that for Fred's ball to hit.

The time it takes for Betty's ball to hit the pins is .... that for Fred's ball to hit.

The time it takes for Betty's ball to hit the pins is .... that for Barney's ball to hit.

The time it takes for Fred's ball to hit the pins is .... that for Wilma's ball to hit.

The time it takes for Wilma's ball to hit the pins is .... that for Betty's ball to hit.

Explanation / Answer

We can deal with these scenarios based on the conservation of energy of the bowling balls

When rolling they will have rotational and translational KE

The overall formula that governs this is mgh = .5mv^2 + .5Iw^2

For a solid ball, I = 2/5mr^2

For a hollow ball I = 2/3mr^2

Also w = v/r

Since KE rotational = .5Iw^2 we have for the solid ball KE = .5(2/5)(mr^2)v^2/r^2 (Note radius cancels - important)

Thus KE rotational = .2mv^2

For the hollow ball KE rotational = (.5)(2/3)(mr^2)(v^2/r^2) (again radius cancels)

Thus KE rotational = .333mv^2

Then for the solid ball we have mgh = .5mv^2 + .2mv^2

Then gh = .7v^2

v = sqrt(1.428gh) = 3.74sqrt(h)

For the hollow ball we have mgh = .5mv^2 + .333mv^2

Then gh = .833v^2

v = sqrt(1.2gh) v = 3.43sqrt(h)

We note that the radius does not matter (nor does mass) so it does not matter that Fred and Barney's balls are scaled down, it only matters if they are hollow or solid

Both solid balls will tie and beat both hollow balls since the velocities of the solid balls are greater. Greater velocity means it gets to the end faster. The hollow balls will tie each other.

Fred and Betty are Solid

Barney and Wilma are Hollow, so

The time it takes for Wilma's ball to hit the pins is equal to that for Barney's ball to hit.

The time it takes for Barney's ball to hit the pins is greater than that for Fred's ball to hit.

The time it takes for Betty's ball to hit the pins is equal to that for Fred's ball to hit.

The time it takes for Betty's ball to hit the pins is less than that for Barney's ball to hit.

The time it takes for Fred's ball to hit the pins is less than that for Wilma's ball to hit.

The time it takes for Wilma's ball to hit the pins is greater than that for Betty's ball to hit.

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