Mechanics simulations and calculators

Where mechanics problems usually go wrong

Almost every error is a sign error or a units error, not a conceptual one. Gravity points down, so it enters the equations as negative 9.81 m/s² whenever you have called up positive. Forgetting that flips the answer’s sign without changing its magnitude, which is why the result still looks entirely plausible.

The value of g deserves a second thought too. It runs from about 9.78 m/s² near the equator to about 9.83 m/s² at the poles, and the conventional standard value is 9.80665 m/s². Quoting a final answer to five significant figures on the back of an assumed 9.81 is a claim about your latitude that you cannot support.

The other recurring trap is mass. It cancels out of free fall entirely, so a heavy ball and a light one fall identically in a vacuum. The moment air resistance enters it stops cancelling, because drag depends on size and speed while weight depends on mass. That is the whole reason a feather and a hammer behave differently on Earth and identically on the Moon.

Energy or forces: which route to take

When a problem gives you a start state and an end state but says nothing about the path between them, use energy. Kinetic and potential energy are state quantities, so the messy middle cancels and a two-line calculation replaces a page of vectors. Setting mgh equal to half mv squared and cancelling the mass is the whole derivation of impact speed: v is the square root of 2gh, so a 1 metre drop lands at 4.4 m/s whatever the object weighs.

Some questions are about a single instant instead: the tension right now, the acceleration at this angle, the force on the tyres in this corner. Those need forces, because energy methods have already averaged that detail away.

The quick test is which words appear in the question. Time, acceleration and force point to a forces problem. Height, speed at two points and energy lost point to an energy problem. A question asking how long or how hard is rarely answerable from energy alone.

The angle is almost never the one in the diagram

Torque uses the sine of the angle between the lever arm and the line of the force, so it is largest at 90 degrees and falls to zero when you push straight along the arm. Work uses the cosine of the angle between the force and the displacement, so it is largest at 0 degrees and zero at 90. The same symbol, θ, appears in both, and swapping sine for cosine converts a maximum into a zero without producing anything that looks wrong.

That is why carrying a bag along a level corridor does no work on the bag. You push up, the bag moves forward, the angle is 90 degrees and the cosine is zero. Your muscles still burn energy holding the tension, but none of it reaches the load.

Launch angles have the same problem. Horizontal velocity is v cos θ and vertical velocity is v sin θ only when θ is measured from the horizontal, which most but not all textbooks do. The check takes a second: at a 90 degree launch the horizontal component must be zero. If your working gives it as v, your angle is measured from the vertical and every component is swapped.

Sanity-check the order of magnitude before the digits

A few anchors catch most arithmetic slips. A 70 kg person weighs about 690 N. A 1 kg mass dropped 1 metre arrives carrying about 9.8 joules. A 1500 kg car at 30 m/s holds about 675 kJ of kinetic energy. A pendulum 1 metre long swings with a period just over 2 seconds. If an answer sits a factor of ten from the nearest anchor, the units are usually the reason.

Then watch what happens when you change an input, because the exponent is a fingerprint. Kinetic energy and centripetal force both go as speed squared, so a 10 percent error in speed becomes a 21 percent error in the result. Period goes as the square root of length, so four times the length doubles the period rather than quadrupling it. If doubling an input moves your answer the wrong way for its exponent, you have dropped a square somewhere.