How Fast Must a Grasshopper Jump to Reach the Other End of a Straw?

[wrobel]

Recently I encountered a funny thing. Consider the following problem from high school course.

A small grasshopper of mass $m$ sits on a tip of a straw. The straw is a thin homogeneous rod of mass $M$ and of length $2l$. The straw lies on a smooth horizontal floor.
What smallest initial speed must have the grasshopper to jump to other tip of the straw?

As a rule, people solve this problem assuming that the grasshopper jumps along the straw. Nevertheless, for some range of $m/M$ this is not true: the smallest speed is attained when the grasshopper jumps sideways from the straw and the straw rotates under him.

:)

 

[mfb]

Jumping along the straw is nice and simple, but jumping to make the straw rotate brings in three coordinates that are all relevant. An interesting case.

 

[osilmag]

I got $v0=\sqrt {2lg}$.

I felt compelled to solve it since I now teach at a high school.

 

[wrobel]

got v0=√2lgv0=\sqrt {2lg}.

Nope. The correct answer contains $m,M$ and for some range of $m/M$ it also contains nontrivial root of equation $const\cdot\varphi=\sin\varphi$

 

[osilmag]

LOL what?

 

[osilmag]

Oh well, you didn't give an angle in your statement.

 

[fresh_42]

the smallest speed is attained when the grasshopper jumps sideways from the straw and the straw rotates under him

Sure? If it can make the straw rotate, then it can make the straw slip underneath it, too, which shortens the necessary length of the jump.

 

[A.T.]

Sure? If it can make the straw rotate, then it can make the straw slip underneath it, too, which shortens the necessary length of the jump.

Obviously. But the OP's point is that for some m/M values jumping non-parallel to the initial straw allows smaller speeds.

 

[A.T.]

2l2l2l. The straw lies on a smooth horizontal floor.

I assume this means friction-less?

Nevertheless, for some range of m/Mm/Mm/M this is not true: the smallest speed is attained when the grasshopper jumps sideways from the straw and the straw rotates under him.

By "sideways" do you mean perpendicular to the straw, or at some non-zero angle? In either case the straw would rotate and translate.

 

[fresh_42]

Obviously. But the OP's point is that for some m/M values jumping non-parallel to the initial straw allows smaller speeds.

I'm not convinced, i.e. I do not see how the question can be solved on the given data. How much speed is needed for a rotation in comparison to a straight slip?

 

[A.T.]

How much speed is needed for a rotation in comparison to a straight slip?

The OP states:

The straw is a thin homogeneous rod

You can get the moment of inertial from that.

 

[wrobel]

I assume this means friction-less?

yes

do you mean perpendicular to the straw, or at some non-zero angle?

I mean some non-zero angle, this angle is a root of equation above

You can get the moment of inertial from that.

exactly
I have not kept my draft, if I remember correctly, for $m\ge 2M$ the grasshopper must jump aside, if $m<2M$ he must jump along the straw

 

[wrobel]

In either case the straw would rotate and translate.

yes it will rotate and translate

 

[A.T.]

if I remember correctly, for m≥2M the grasshopper must jump aside,

Interesting. Do you remember if the horizontal jump angle approaches some value like 90° when m/M approaches infinity?

 

[wrobel]

pardon, $\varphi$ is an angle of straw's turn at the moment when the grasshopper landed on the tip.
I will restore my solution and put it here

 

[wrobel]

here it is