Problem of the Week 1243

Weight Loss Through Juggling

You are driving a truck carrying 15 heavy rocks, all weighing the same amount. Each rock is mounted on top of one of 15 launchers. Each launcher can throw its rock into the air and then catch it again (air resistance is ignored). When a launcher is activated, it pushes up on the rock for one second, the rock is then airborne for two seconds, and when the launcher catches the rock, it again pushes up on the rock for one second to bring it back to rest.

By Newton's third law, during the second when the rock is being thrown and the second when it is being caught, not only does the launcher push up on the rock, but the rock also pushes down on the launcher. The result is that, in effect, when a launcher is activated, its rock weighs twice its normal weight for one second, then nothing for two seconds, and then twice its weight for one second before returning to its normal weight.

You come to a bridge that can support the weight of the truck, you, all the launchers, and 14 rocks, but not 15 rocks. Fortunately, it takes only two seconds for the truck to cross the bridge. One second before reaching the bridge, you activate one of the launchers. During the last second before you reach the bridge, the rock that is being launched weighs twice its normal weight, but you are not yet on the bridge so this weight increase is not a problem. During the two seconds when you are on the bridge, the rock is airborne so the load does not exceed the bridge's weight limit and you make it to the other side. During the first second on the other side, the launcher catches the rock, again causing a harmless increase in the weight of the rock.

You are pleased with your cleverness in getting across this bridge with all the rocks, until you see a sign warning you that another bridge is ahead. This bridge has the same weight limit as the last one, but it is longer: it will take you six seconds to get across.

How can you get everything across without exceeding the weight limit of the bridge?

Source: Dan Velleman (Amherst College)