Projectile motion ball question

[Kristin_Z]

A ball moving with speed v rolls off a shelf of height h and strikes the floor below a distance x₁ from the edge. A second ball moving with the same speed rolls of a self of height 2h and strikes the floor a distance x₂ away from the edge. Determine the ratio of distances x₁/x₂.
$$\Delta$$x=V_xt
$$\Delta$$y=vt +1/2st²
The horizontal motion of the ball is constant and, therefore, not affected by the height from which it rolls. But I cannot figure out how to formulate the ratio of the distances. I know that time relates the two equations of motion and that because the ball is rolling it has no initial vertical velocity and that since it is free fall the acceleration would be the same for the two equations. I just can't seem to get the equation to make sense when I write it out. Any help would be greatly appreciated. Thanks!

 

[LowlyPion]

A ball moving with speed v rolls off a shelf of height h and strikes the floor below a distance x₁ from the edge. A second ball moving with the same speed rolls of a self of height 2h and strikes the floor a distance x₂ away from the edge. Determine the ratio of distances x₁/x₂. The horizontal motion of the ball is constant and, therefore, not affected by the height from which it rolls. But I cannot figure out how to formulate the ratio of the distances. I know that time relates the two equations of motion and that because the ball is rolling it has no initial vertical velocity and that since it is free fall the acceleration would be the same for the two equations. I just can't seem to get the equation to make sense when I write it out. Any help would be greatly appreciated. Thanks!

Well I can see you are almost grasping it.

The horizontal motion is constant, that's true, but what affects the distance at which it lands?

d1 = V*t1
d2 = V*t2

So ... the ratio of the distances then will be in the same ratio as the ratio of the times won't it?

And now is there some way to relate the ratio of the times using the heights?

 

[Kristin_Z]

I think I have it now.

2y = 1/2 at₁² divided by y = 1/2 at₂²

gives me 2= t₁²t₂²

so t₁t₂ =square root of 1/2.

Thanks for your help, still not entirely sure I understand what I did. I guess that's what practice is for.

 

[rl.bhat]

I think I have it now.

so t₁t₂ =square root of 1/2.
.

Check this step.

 

[LowlyPion]

I think I have it now.

2y = 1/2 at₁² divided by y = 1/2 at₂²

gives me 2= t₁²t₂²

so t₁t₂ =square root of 1/2.

Thanks for your help, still not entirely sure I understand what I did. I guess that's what practice is for.

I think you really meant
2= t₁²/t₂²

Which yields the ratio

t₁/t₂ = √2

But I would note the original statement has y2 being 2*y1, so I think you have your subscripts reversed.

The higher ball would fall farther from the base by a factor of √2 is the sense of you should be grasping.