It also means that the bowl curves in on itself so that if the marble leaves the bowl it will perform projectile motion across the top of the bowl. Capisce?
Note that the range of a projectile is given by the formula where v is the initial speed and is the angle of projection (measured from the horizontal).
You therefore require to be greater than the distance across the bowl (otherwise the marble will land in the bowl. Capisce?).
Let be the angle between the line across the top of the bowl and the radius to the top edge of the bowl (where the marble is going to project from). Then, using simple trigonometry on the obvious 90 degree triangle, you have the following:
1. The distance across the top of the bowl is equal to .
3. The angle of projection of the marble (measured from the horizontal line across the top of the bowl is . (This comes from the fact that the marble leaves the bowl at a tangent to the bowl and the fact that the tangent is 90 degrees to the radius).
I'd draw a diagram but it's good for you to draw one and nut this out ...... (that plus the fact that I'm just plain lazy).
To summarise then:
The marble leaves the bowl with initial velocity v and angle of projection to the horizontal. It undergoes projectile motion and you require the range of the marble to be greater than the distance across the bowl.
Mathematically, the requirement is:
So the value of is needed to finish things off. This can be got via conservation of energy .....
Take G.(ravitational) P.(otential) E.(nergy) of marble at bottom of bowl equal to zero. Then:
(K.E. + G.P.E) of marble at top of bowl = K.E. of marble at bottom of bowl
Substitute this result into (A):