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Two-Dimensional Problems? No Problem.

When you’re dealing with another dimension, don’t make the problem too difficult for yourself. You don’t have to relearn the basics of momentum or kinematics all over again. If you’re having trouble with a problem involving 2+ dimensions, my very best advice is for you to imagine one observer for each dimension. The laws of physics should hold for all observers. There is a base set of equations or relationships you’ll need to solve the problem, but you don’t need a new set of equations for new dimensions. Though new equations may ensue by having another dimension, the equations for one dimension will be enough to solve the problem if you consider multiple observers watching the same event from different vantage points (2 observers for 2D, etc.). Though both observers don’t have a complete understanding of what’s going on from their own point of view, these witnesses can use the data that they both agree on to figure out what the unknown is. So, how does this work? Give me an example.

In the conservation of momentum problem in 2D, consider one person looking down the x-axis and one person looking down the y-axis. Momentum should be conserved in both of their views, even though both observers don’t have a complete understanding of the interaction. So, both observers see that momentum before and after the collision are the same (momentum is conserved).

In projectile motion problems, 2D is simple if one observer has full view of the y-axis and the other of the x-axis. In other words, Jane is sitting behind the batter and she only sees the ball go up at an initial velocity, peaking at a certain height and then dropping to the ground. Hans, is up above in a blimp looking down (or watching the shadow of the ball) and he sees his own initial velocity and, with no wind resistance, the ball seems to travel at that velocity until it hits the ground. They both may measure differing initial velocities and traveling path distances, but they will measure the same time. Of course! If you are given an unknown (i.e. if Hans didn’t determine the distance traveled), you should be able to get the equations Jane uses and link it to Han’s equations by the fact that the measured time ‘t’ is the same. The key is that Hans and Jane both use the same equation “cheat sheet;” they just choose to use different equations because Jane sees the effect of gravitational acceleration and Hans doesn’t.