Optimal Path for Honeybee to Reach Honey Drop in Minimum Time - No Gravity

[PhysicsKT]

Homework Statement

A honeybee is flying parallel to tabletop at height

with velocity

.With its wings it can acquire a maximum acceleration of

. At an instant when honeybee is vertically above honey drop on tabletop and decides to reach the honey drop. Neglect reaction time of honeybee and there is no gravity. Find minimum time in which honeybee can reach honey drop.

2. The attempt at a solution

Let us work in the frame moving with the constant velocity

, such that now the honeybee is simply moving with

and the honey drop moves in the opposite direction (to the initial velocity of the honeybee) with a constant velocity

. The honeybee obviously wants to travel the least distance in order to catch the honey drop, so should compute the point it will catch the honey drop if it travels on a straight line towards that point. Let the honey drop's coordinates at a certain time defined

be

, and the honeybee be at

at this time. After some time

, the coordinates of the honey drop would be

. Now, obviously

The solution of this equation would be at

, that is, when the honey drop is finally caught by the honey bee. Using the well-known quadratic formula for the equation written above,

My question is: is this wrong? The answer matches, but an acquaintance comes and asks me, why did you assume linear acceleration? Why do you think honeybee only wants to travel the least distance? Because the problem says: find the minimum time...

Please help me asap.

 

[PeroK]

My question is: is this wrong? The answer matches, but an acquaintance comes and asks me, why did you assume linear acceleration? Why do you think honeybee only wants to travel the least distance? Because the problem says: find the minimum time...

Please help me asap.

It's logical that if you start at a point and need to reach a point on a given line, then a straight line will minimise the distance. If you travel further, along a curve say, then you need a greater average speed to get there in the same time. And, linear acceleration maximises your speed, as well. An acceleration component that changes direction does not increase your speed.

 

[DrClaude]

@PhysicsKT, please check out https://www.physicsforums.com/help/latexhelp/

Typing the equations directly into the text, instead of posting them as images, makes the post easier to read and to quote.

 

[PhysicsKT]

It's logical that if you start at a point and need to reach a point on a given line, then a straight line will minimise the distance. If you travel further, along a curve say, then you need a greater average speed to get there in the same time. And, linear acceleration maximises your speed, as well. An acceleration component that changes direction does not increase your speed.

But what we want is to minimise time, not distance.

 

[PeroK]

But what we want is to minimise time, not distance.

If you minimise distance and maximise speed, then you must minimise time. You can't go further and slower in less time.

 

[PhysicsKT]

Thanks. Got it!