Here's a brainteaser I thought up recently.
A cannon fires a professional baseball player 60 degrees from the ground at 80 m/s. The cannon faces a pipe 245 meters away. The pipe is stuck into the ground pointing straight into the air and has a diameter slightly large than a baseball. Could the pitcher in midair throw a baseball so that it goes into the pipe but does not touch the sides? Explain why or why not.
>>7682221
... Woah.
>>7682261
What?
What is the max speed the pitcher can through?
>>7682276
Good question, 100 miles per hour
Have the mods already given up on this board?
>>7682294
If you're implying this is a homework question, it's not. I made it up this morning.
>>7682296
pretty specific.
>>7682298
What is? The question? You don't need much math to figure out the answer. It's more of a conceptual problem. I'll post the answer in an hour if no one figures it out.
>>7682301
If the baseball player managed to throw the ball at the exact right angle so that it's horizontal component of movement was reduced to zero, it would be falling straight down and thus it would be able to go into the pipe without touching the sides.
The following calculations assume no air resistance.
the baseball player's horizontal motion is [math]80\cos(60$^{\circ}$) = 40[/math]
100 mph = 44.4 m/s
so the baseball player would have to throw the ball so that the horizontal component of his throw (at 44.4 m/s) is 40 m/s.
[math]\arccos(40/44.4) = 25.8$^{\circ}$[/math]
so he'd have to throw the ball about 25.8 degrees below horizontal towards the cannon.
>>7682320
just ignore those $$
>>7682221
yes. simple two dimensional kinetics. all the pitcher needs to do is give the ball a trajectory so the horizontal component of the ball once launched is zero. the vertical component does not matter as gravity can do the work.
Further, 80 m/s is about 180 mile an hour. if the pitcher can throw the ball at a maximum of 100 miles and hour he can achieve this feat. given that the cosine of 60 degrees is .5 he can achieve this feat at 90 miles an hour as well. The angle is not important as the baseball player would likely die as soon as he hit the ground.
>>7682320
You're correct but your second calculation was unnecessary. He just needs to throw the ball horizontally at 40 m/s. He doesn't have to throw the ball at max speed at an odd angle.
>>7682320
The angle of the throw is not relevant. The player could throw the ball horizontally and still arrest its forward motion. (It might continue upward before falling into the pipe.) The critical question is whether or not he is capable of throwing a ball that hard. As you've shown, he just has to throw at 40 m/s, which is about 90 mph.
>>7682331
That's very true.
Ah well I just thought it was an interesting problem because the idea of the pitcher throwing the ball backwards in order to get it to just fall into the pipe is pretty un-intuitive. Most people would think that he needs to aim the ball into the pipe but then the conserved momentum of the ball would make it hit the walls of the pipe.
>>7682221
Yes. The pitcher can throw the ball in a parabolic arc steep enough that when it intercepts the pipe at a distance of 245m, its slope is low enough that it does not move far enough horizontally to touch the sides before it has reached the bottom of the pipe. If the pipe is infinitely long, he has to account for friction with the air to stop horizontal movement just over the pipe.
