Thread: Projectile motion: aiming at a monkey

1) Suppose the zookeeper must shoot the banana from the banana cannon to the monkey who hangs from the limb of a tree. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon?


The answer is to aim directly on the monkey, but I don't understand why!
I can only understand the special case when the cannon is at the same height as the monkey and is shot horizontally aiming at the monkey, they will drop by the same distance because horizontal and vertical components of a projectile motion are independent.

However, I don't understand the case above. Can somebody briefly explain the reason to me? Thanks a lot!

well its probably due to the gravitational force .. like for example in dis case .. wen the shot is fired aiming directly at the monkey .. the gravitational force will act on the banana n reduce its height of elevation until it reaches the monkey .. so wen the monkey is in mid-air, the banana will probably reach the same height at the same time ...
well dis is wat i think .. im not too sure tho .. mayb sum1 els cud give a better explanation ..

Well the proper way to do this is with the mathematics.

let x be the horizontal distance of the tree from the banana cannon
let y be the starting height of the monkey
let x1 be the horizontal distance of the banana from the cannon
let y1 be the vertical distance of the banana from the cannon
let x2 be the horizontal distance of the monkey from the cannon
let y2 be the vertical distance of the monkey from the cannon
let a be the angle from the horizontal that the cannon is pointed
let v be the velocity of the banana when it first comes out of the cannon
Then
x1 = (v cos a) t
y1 = (v sin a) t - (4.9 m/s^2) t^2
x2 = x
y2 = y - (4.9 m/s^2) t^2

If we want the banana and the monkey to meet then we want x1=x2 and y1=y2.

x1= (v cos a) t = x = x2
y1= (v sin a) t - (4.9 m/s^2) t^2 = y - (4.9 m/s^2) t^2 = y2

In the 2nd of the 2 highlighted equations we can add (4.9 m/s^2) t^2 to both sides to get the following two equations.

x = (v cos a) t
y = (v sin a) t

now solve for a

cos a = x/(vt)
sin a = y/(vt)

or dividing the second by the first

tan a = y/x

But this means that you point directly at the monkey.

In retrospect looking back at the math we see that this happened because the action of gravity on the banana and the money were exactly the same and canceled each other out.

Thanks a lot!

mitchell wouldn't it be easier to say that its because of the curved trajectory of the bannana.

Originally Posted by wallaby

mitchell wouldn't it be easier to say that its because of the curved trajectory of the bannana.

Intuition is a great tool in science but it can mislead. Aristotle for example was a great philosopher and "scientist" of his time (admired for centuries after his death) but his use of intuition in physics shows just how badly it can mislead you in physics. So it is only when intuition and the mathematics agree that you can really be sure that you have it right. But when intuition gives you correct answer it tends to make the mathematics a lot easier.

The calumny that Aristotle did not believe in the experimental method is one that should be thoroughly debunked. It arises out of three things
a) The loss of most of Aristotles work.
b) Gross misinterpretation by later scholars.
c) That self aggrandising bastard Bacon.
I shall start a thread on it at some point.

Originally Posted by Ophiolite

The calumny that Aristotle did not believe in the experimental method is one that should be thoroughly debunked. It arises out of three things
a) The loss of most of Aristotles work.
b) Gross misinterpretation by later scholars.
c) That self aggrandising bastard Bacon.
I shall start a thread on it at some point.

Ohh... Ophiolite I agree with you 100%. I am a great admirer of Aristotle, am sorrowful at the misunderstandings about him, and I said nothing about Aristotle not believing in the experimenal method!!!!! But this just goes even more to prove my point of how intuition can mislead, because Aristotle was not stupid or unscientific but he made those mistakes anyway.

Your post seemed to imply that he only used intution, untested by experiment. I am pleased to see you share my positive view of Aristotle.

Originally Posted by Ophiolite

Your post seemed to imply that he only used intution, untested by experiment. I am pleased to see you share my positive view of Aristotle.

Looking back I guess it was putting scientist in quotes, but all I meant by that is that science proper really began at a later time. It was an unfortunate unintended implication, because I certainly think he was more of a scientist than anyone else in pretty sizable period of time. His painstaking attention to detail, observations in biology and anatomy, and in-depth logical analysis was all way ahead of his time.

Originally Posted by kingwinner

1) Suppose the zookeeper must shoot the banana from the banana cannon to the monkey who hangs from the limb of a tree. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon?


The answer is to aim directly on the monkey, but I don't understand why!
I can only understand the special case when the cannon is at the same height as the monkey and is shot horizontally aiming at the monkey, they will drop by the same distance because horizontal and vertical components of a projectile motion are independent.

However, I don't understand the case above. Can somebody briefly explain the reason to me? Thanks a lot!

put simply: the cannon would have to be horizontal with the monkey for this to be true. than they would fall at the same rate. if the cannon was aimed upward, than it would have more vertical motion and would go over the monkey.

Originally Posted by chamilton333

put simply: the cannon would have to be horizontal with the monkey for this to be true. than they would fall at the same rate. if the cannon was aimed upward, than it would have more vertical motion and would go over the monkey.

This is incorrect. This person should read the whole thread more carefully.

Wow, Mitchell, that's some awesome mathamatics! It's one of those problems I find hard to do, but easy to understand. You must be a good teacher; now I'm jealous of your students. :P