Suppose two identical marbles (A and B) are rolled off of a horizontal tabletop, after which they undergo free-fall movement throughout their respective flights toward the ground. Marble A is seen to be rolling at a slower speed on the tabletop (as compared to B) upon reaching the edge of the table. In the space below, determine and explain (i) which marble, if any, will be in flight for a shorter amount of time, and (ii) which marble, if any, will hit the ground at a faster overall speed?

Question (i): "Neither of the two marbles will hit the ground in a shorter time"

Question (ii): "Marble B will hit the ground with a faster overall speed".

Step-by-step explanation

Solution:

Data:

Objects A and B are identical in mass and size. Also for the calculation we will consider that they are point masses.

The height from which objects fall is the same for both: ?hA?=hB?=h?

The horizontal speed of A is less than the horizontal speed of B: ?vAx?<vBx??

Question (i):

The vertical position of both objects is related to the vertical component of the initial velocity and to the fall time through the following equation:

?yf?=y0?+v0y?⋅t−21?g⋅t20=(y0?−yf?)+v0y?⋅t−21?g⋅t2?

Note that:

The vertical component of the initial velocity for both objects is zero because it is a free fall:

?v(0y)?=v(0yA)?=v(0yB)?=0?

And: ?y0?−yf?=h?

Therefore, by applying this equation to both objects, the following equations are obtained:

?h−21?gtA2?=0 → tA?=g2h??h−21?gtB2?=0 → tB?=g2h???

Therefore, the fall times of both bodies are the same, or, "neither of the two marbles will hit the ground in a shorter time"

Question (ii):

For marble A, its total final speed is given by:

?vfA?=vfxA2?+vfyA2???

With :

?vfxA?=vAx?vfyA?=gtA?=gg2h???

Then:

?vfA?=(vAx?)2+(gg2h??)2?=vAx2?+2gh??

In the same way, for marble B you have to:

?vfB?=vfxB2?+vfyB2???

With:

?vfxB?=vBx?vfyB?=gtB?=gg2h???

Then:

?vfB?=(vBx?)2+(gg2h??)2?=vBx2?+2gh??

Finally note that the final values ??are as follows:

?vfA?=vAx2?+2gh?vfB?=vBx2?+2gh??

and since the statement states that ?vAx?<vBx?? , then it is true that:

?vfA?<vfB??

Therefore, "marble B will hit the ground with a faster overall speed".