Really, 50 feet? If this were your answer to a physics problem, you would probably want to check your math.
Welcome back! In this series of posts, let’s explore a few methods you can use to solve physics problems more easily, quickly, and accurately. In my work as a physics and math tutor, I often notice that good problem-solving habits, like good study or eating habits, can make all the difference. In our last post, we discussed the importance of thinking about the answer you expect to find when you first approach a problem. In the next few posts, let’s look at a few questions that you should ask yourself once you get your answer.
Is the physical quantity reasonable?
When you come to your answer, you should remember that the number isn’t simply a series of digits to be copied from your calculator. It’s a physical quantity; it represents an object moving at some velocity, or falling from some height, or possessing some amount of charge.
Accordingly, you have a wealth of accumulated physical knowledge that might enable you to judge whether your answer is reasonable. Let’s say you are asked to find the velocity of an Olympic skier at the bottom of a snow-covered ramp. Is it reasonable for that skier to be moving 200 m/s? Could he traverse two football fields in the time it takes you to say “touchdown?” Olympian stature aside, the answer is “probably not.” Although not an ironclad indictment of your answer, such a stretch of your imagination suggests that you might want to go back and check your work.
Use what you know
While using this approach, use familiar measurements and don’t be afraid to be approximate. One meter is about three feet and one kilogram is about two pounds. Approximate lengths of football fields (say 100 yards is about 100 m), heights and weights of people (say 6 ft and 150 lbs are about 2 m and 75 kg1), or other commonly known measurements are your friends here, as are rounded numbers. If you calculate that a certain decelerating car has a mass of 1,600 kg, you might ask yourself if you believe a car might weigh as much as 20 160-lb people. It seems reasonable to me.
This approach will not always work. You might not know whether 200 μC is a reasonable charge for a metal ball to have or whether a satellite might orbit the earth with a velocity of 1,550 m/s. You can, however, decide that an office building might be 32m tall – 30 m is about 90 ft, which might be equal to 9 stories if two 5-ft tall people can be stacked in one story – but not 1,210 m tall, or you can decide that a car might cover 18 m – 20 m is about 60 feet, or ten body lengths – in a second, but not 180 m.
Helpful check, not ironclad proof
Einstein said that no amount of experimentation could ever prove him right, but a single experiment could prove him wrong. In a similar spirit, this kind of order-of-magnitude sanity check can only flag a suspicious answer. It can’t serve as proof that a sensible answer is correct. The fact that a car could reasonable drive down a street with a velocity of 23 m/s does not imply that it must do so.
Inversely, failing this sanity check can only imply that an answer is suspicious, not prove that it is wrong. The fact that a car would not normally drive down a street with a velocity of 4.5 m/s does not mean that it could not do so.
Essentially, I am encouraging you to stop whenever possible and ask yourself whether what you are seeing makes sense. This mindset will serve you well in your physics classes and, really, would not make a bad approach to life more generally. Carry with you the caveats that wrong answers often seem perfectly sensible and that improbable answers sometimes turn out to be correct, but, as a scientist in training, it does you no harm to ask the question.
1 Yes, these conversions are approximate, and we should be mindful that a pound is a unit of force and a kilogram is a unit of mass, but we just want to establish a few easily remembered and intuitively accessible references. Errors of 20% or less don’t matter much if we are just asking ourselves whether we have the correct order of magnitude, and few of us have ever experienced a gravitational acceleration other than 9.8 m/s2, which gives us the standard conversion between pounds of weight and kilograms of mass.