Puzzles/How do you ... ?/Crossing the Bridge/Solution
The puzzle states that the man currently WEIGHS 98lbs. There is no mention of his mass, and so we must therefore assume that if "the man weighs exactly 98 pounds", and he is holding the three 1 pound apples, as is stated, the 98 pounds INCLUDES the 3 apples already. Therefore to cross the bridge, the man runs, skips, jumps or simply strolls across the bridge, with no fear of collapse. He may, however, collapse at seeing the price of an apple that weighs a pound...
He juggles the apples across.
- Since he has to throw the apple to juggle it, he would end up bearing more than the force of holding the apple staticly. So I'm not sure this would work.
- This should work as long as he doesn't throw too hard. To be sure, he could hold only one at any time...
- Agreed with above. He is effectively "juggling" one apple if he staggers the juggling. He could juggle as many as he likes across the bridge.
(Sorry for shouting, the laws of physics made me do it!)
A juggler who weighs 98 pounds juggling 3 apples weighing 1 pound each imparts a mean force of 101 pounds on the bridge. Mean force refers to the average force over time.
Objects thrown up will be at the same speed falling at the height they were thrown from. Consider that at any given time she has one apple in the hand and two in the air. Assume that a throwing velocity of V is needed to keep these apples in the air. Since there are three apples, each apple spends two thirds of its time in the air, and one third in her hand. While in the air, the apples accelerate down at g changing velocity from V to -V. In the hand the apples need to accelerate in the opposite direction, for a speed change from -V to V.
But they have half the freefall time to do this (two apples in air, one in hand at any instant of time!) Thus, while in her hand, they are accelerating at 2g upwards. They are doing this in a -1g gravitational field.
Sir Isaac says that the resultant force that each apple imparts on the juggler is 3 pounds downwards. Hence, the juggler imparts 98 + 3 = 101 pounds on the bridge.
However, all is not lost! The way to cross the bridge without exceeding the 100 lb limit is as follows:
Hold apples in your raised hand. Run as fast as you can to cross the bridge, but at the instant you get to the bridge bring your arms down in one continuous motion (accelerating the apples down at g/3). You must not finish this until you have crossed the bridge!
If your arms are 2 ft long, you have 4 feet to lower the apples. Since g=32 ft/sec/sec and s=0.5*a*t*t, then you must lower the three apples at g/3 to reduce your load by 1 lb. So we have: t*t = 4/(0.5*32/3) or approximately 0.75 sec*sec, thus t is approximately 0.85 seconds.
Therefore, if you can run fast enough to cross the bridge in under 850 milliseconds and can move your arms in the motion described - at just the right time - you can cross the bridge.
We assumed your arms have negligible mass, in reality you get more time, because you are also accelerating the mass of your arms. That calculation is left as an exercise to the reader. (mce)
He could also cross the bridge 3 times carrying each apple every time.
(Or, he could bowl one across, assuming he doesn't mind risking losing an apple to the river.)
He eats all three apples, digests them, loses a pound by excreting and urinating then walks over
- He left the digested apples so he didn't cross the bridge WITH the apples
He sacrifices a part of his body weighing one pound
He prays that the verticle g's his body creates doesn't lead to his demise, and he hopes the engineer who calculated the weight limit left a margin for error.
He gets a hair cut.
He runs really fast across the bridge, and lets the centripetal force kick in and take some of that weight off.
He waits until there is a full moon. With the additional gravitational pull, he can successfully cross the bridge.