Physics problem solving is mostly checking your answer before you know it's wrong

In chess, the commonest way to lose is not a bad plan. It's a good-looking move you didn't check. You see a clean tactic, play it, and only after your opponent replies do you notice the one square you forgot to look at. Strong players aren't smarter at finding moves. They have a habit of one extra look before letting go of the piece.

Physics problem solving has the same disease. I derive an expression, circle it, move on. And later I find out in the answer key that I was off by a factor of two, or that I'd squared the wrong thing. For a long time I thought the fix was "be more careful". That never works. The fix is a blunder check, a small routine that takes twenty seconds. I have three.

Check 1: dimensions

Every term you add must have the same dimensions, and your final answer must have the dimensions of what was asked. For a pendulum:

$T = 2\pi\sqrt{\frac{L}{g}}$

$\dfrac{L}{g}$ has dimensions $\dfrac{\text{m}}{\text{m s}^{-2}} = \text{s}^2$, so the square root is in seconds. ✓. If you had misremembered it as $\sqrt{g/L}$, you'd get $\text{s}^{-1}$ and know immediately that it's wrong.

Dimensions catch: adding unlike quantities, forgetting to square or take a root, a $g$ that fell to the wrong side of a fraction. They don't catch: pure numbers like 2, $\pi$, $\tfrac12$ and trigonometric factors.

Check 2: limiting cases

Push each variable to an extreme where you already know the answer. This is the most powerful check and nobody teaches it.

Atwood machine, two masses $m_1 > m_2$ over a light frictionless pulley. A derivation gives

$a = g\,\frac{m_1 - m_2}{m_1 + m_2}, \qquad T = \frac{2m_1m_2}{m_1+m_2}\,g$

Now test it:

• $m_1 = m_2$: $a = 0$. Balanced, nothing moves. ✓
• $m_2 \to 0$: $a \to g$. $m_1$ is in free fall, nothing pulling back. ✓
• $m_2 \to 0$: $T \to 0$. A free-falling mass has no rope tension. ✓
• $m_1 = m_2 = m$: $T = mg$. The rope holds up one weight. ✓
• Swap the labels $m_1 \leftrightarrow m_2$: $a$ changes sign. Motion reverses. ✓

Five checks in under a minute, and I haven't looked at the answer key.

Here's one that caught a real mistake. In a mock I wrote the projectile range as

$R = \frac{u^2\sin\theta}{g}$

Dimensions: $u^2/g$ is a length. ✓ Looks fine. Now limits: at $\theta = 0$ we get $R = 0$, correct (no height, lands immediately). But at $\theta = 90^\circ$ it gives $R = u^2/g$, and a stone thrown straight up lands at its starting point, so $R$ should be $0$. ✗. The correct formula has $\sin 2\theta$:

$R = \frac{u^2\sin 2\theta}{g}$

which is $0$ at both ends and has a maximum at $45^\circ$. That wrong answer would have passed my dimension check. The limit check killed it.

Check 3: symmetry and conservation

Ask what the problem should not depend on.

• A block slides down a smooth track of unknown shape from height $h$. Speed at the bottom? You don't need the shape. Energy conservation gives $v = \sqrt{2gh}$ for any smooth path. If your answer depends on the path, you made it harder than it is.
• Two identical masses collide head-on elastically. They exchange velocities. If your answer doesn't do that, something's off.

I think of conservation laws as endgame tables in chess. In an endgame with few pieces, the result is already known, so you skip the whole middle and read the answer. When a conservation law applies, you skip the differential equation and read the answer.

Check 4 (the free one): is the number believable?

If someone jumps with a launch speed of 3 m/s, the height is $\dfrac{u^2}{2g} = \dfrac{9}{20} \approx 0.45$ m. That's a plausible human jump. If my answer is 45 m, I've lost a factor of 100 somewhere.

What it looks like in practice

Check	Time	Catches	Misses
Dimensions	10 s	wrong powers, mixed-up quantities	numerical factors, $\sin$ vs $\cos$
Limiting cases	20–30 s	wrong trig factors, wrong signs, wrong denominators	errors that vanish at the limits you tried
Symmetry / conservation	15 s	answers that depend on things they shouldn't	not much, but needs insight
Sanity of size	5 s	factors of 10, 100	small errors

None of them proves your answer is right. They only prove it isn't wrong in these ways. But "not wrong in four ways" is a long distance from "I circled it and hoped".

I didn't expect this habit to change my scores much. It did. Mostly because my mistakes were rarely about not knowing physics. They were about not looking once more. In chess I play slower now too, and I lose fewer pieces to my own hurry.