Beetle harvest

Classical Mechanics Level 3

A scout from an ant colony (colony A) finds a dead beetle of mass M M a distance d A d_A from his colony. At the same time, a scout from another colony (colony B) who is distance d B d_B from his colony, also finds the beetle. If ants from colony B can walk at speed v B v_B , and workers from A A can each carry the mass δ m \delta m of ant flesh at any given time, what is the slowest speed (in m/s) at which workers from colony A walk so that workers from colony B never catch up and steal the beetle pieces?

Details

  • If an ant from colony B catches an ant from colony A, it will kill the ant from colony A.
  • If an ant from colony A makes it to colony A with beetle flesh, he is safe from workers from colony B.
  • The scouts both start out at the beetle.
  • d A = 3 d_A=3 m
  • d B = 5 d_B=5 m
  • v B = 5 v_B=5 m/s
  • M = 250 M=250 g
  • δ m = 0.5 \delta m=0.5 g


The answer is 3.46154.

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Josh Silverman by Josh Silverman

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1 solution

Adrian Peasey
Jun 25, 2015

To get back safe the ants from A have to get the beetle home before the ants from B can catch up, i.e. they have to get back quicker than the ants from B can reach colony A.

The scout from A has to return to the colony, then bring workers back to the beetle and home again, so they travel d A d_A 3 times. Meaning the total distance they travel is x A = 3 d A = 3 × 3 = 9 x_A=3d_A=3\times3=9 .

The scout from B has to return to its colony and bring workers to colony A, so they travel d B d_B twice and d A d_A once. Meaning the total distance they travel is x B = d A + 2 d B = 3 + 2 × 5 = 13 x_B=d_A+2d_B=3+2\times5=13 .

Since t = x v t=\frac{x}{v} , and we know the ants from A A have to be quicker we now have t A ≤ t B ⇒ x A v A ≤ x B v B ⇒ v A ≥ x A v B x B t_A\leq t_B\Rightarrow\frac{x_A}{v_A}\leq\frac{x_B}{v_B}\Rightarrow v_A\geq\frac{x_A v_B}{x_B} .

Substituting in the numbers we get v A ≥ 9 × 5 13 = 3.462 v_A\geq\frac{9\times5}{13}=\boxed{3.462} .

The mass is irrelevant since there is no limit given for the number of ants,

6 Helpful 3 Interesting 3 Brilliant 4 Confused

I solved the problem this way too, but why can't the scout from B just kill the scout from A? He can outrun the scout if v_A=3.462. I think the problem should state that the scouts first must return to their colonies before B decides to kill ants from A.

Alex Wang - 5 years, 11 months ago

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Yes the question should have stated that

Rudrayan Kundu - 2 years, 9 months ago

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