Math Model for Archer Fish Spitting Water at Insect

[atomant]

I came across a question involving an archer fish and an insect that has me puzled. The question gives that an archer fish spots an insect dangling on a branch of a tree. The fish can attack by spitting water at it's prey. But it just so happens that just as this fish spits out the water, the insect is startled and drops from the branch. Now the question is what happens to the water that was spat out?..does it miss the insect?, or does it hit regardless?.
The answer given was that it hits the insect regardless of the drop. I do not however understand why this is so?. I have tried to come up with an equation to explain this but no luck so far. All I have so far is that there is an object (the insect) that is falling at a constant rate of g, another object (the water from the fish) traveling towards it with a speed v and the fact that at time t the two objects collide. Is there any way to show this through an equation?

 

[topsquark]

I came across a question involving an archer fish and an insect that has me puzled. The question gives that an archer fish spots an insect dangling on a branch of a tree. The fish can attack by spitting water at it's prey. But it just so happens that just as this fish spits out the water, the insect is startled and drops from the branch. Now the question is what happens to the water that was spat out?..does it miss the insect?, or does it hit regardless?.
The answer given was that it hits the insect regardless of the drop. I do not however understand why this is so?. I have tried to come up with an equation to explain this but no luck so far. All I have so far is that there is an object (the insect) that is falling at a constant rate of g, another object (the water from the fish) traveling towards it with a speed v and the fact that at time t the two objects collide. Is there any way to show this through an equation?

This is similar to the "shoot the monkey" (or for those that like a more animal friendly problem, "shoot the can") problem. Don't think of the water as a stream, but a projectile. As the insect falls, it is accelerated by gravity. As the water moves, it is accelerated by the same amount from where it would be if it traveled in a straight line. (Sorry, I can't do the picture yet!)

In terms of equations, what you need to do to set this up is to write an equation that says where the insect is at a given time. Do the same for a "projectile" (your steam of water) from the fish. What you want to show is that when the projectile gets to the horizontal (x) displacement that the insect is at, that both the insect and projectile have the same vertical (y) displacement.

It isn't a particularly hard problem, but you do need to keep careful track of all the variables involved.

-Dan