Physics help that checks your work as you write

alai is a notes app for iPad and the web with Ally, an adaptive AI tutor that teaches alongside you. A physics solution mixes a sketch, an equation, numbers with units and some algebra, and a slip in any of them carries through to the answer.

Where physics goes wrong on paper

Kinematics and signs

The constant-acceleration equations, v = v₀ + at, Δx = v₀t + ½at² and v² = v₀² + 2aΔx, only work when every quantity follows the same sign convention. Pick a positive direction first and give each value its sign.

Throw a ball straight up at 12 m/s and take up as positive. Gravity points down, so a = −9.8 m/s². After 2.0 s, v = 12 + (−9.8)(2.0) = −7.6 m/s, and the minus sign tells you the ball is on its way down. Using +9.8 instead gives 31.6 m/s, a ball that speeds up as it rises. At the top of the flight the velocity is zero, but the acceleration is still −9.8 m/s².

Braking works the same way. A car going 25 m/s that stops in 50 m gives 0 = 25² + 2a(50), so a = −6.25 m/s². The acceleration is negative because it points against the motion.

Forces and free-body diagrams

A free-body diagram shows every force acting on one object, and only those. OpenStax's University Physics textbook says to leave the net force and the acceleration out of the diagram (University Physics Volume 1, section 5.7). In a two-year study, students who drew their free-body diagrams correctly were significantly more successful at solving the problems (Rosengrant, Van Heuvelen and Etkina, 2009).

• Using g for the weight. A rope pulls a 2.0 kg box straight up with a tension of 30 N. With up as positive, T − mg = ma gives 30 − 19.6 = 2.0a, so a = 5.2 m/s². Writing 30 − 9.8 = 2.0a puts g where the weight, mg, belongs.
• Drawing an arrow that isn't a force. The product ma is the result of the forces, so it doesn't get its own arrow. Centripetal force works the same way: it's the name for the net inward force, so it comes from arrows you've already drawn.
• Assuming the normal force is mg. On a ramp at 30°, the normal force on a 5.0 kg block balances only the part of the weight pressing into the ramp, mg cos 30°, which is about 42 N. The full weight is 49 N.

Units

Convert before you calculate. A cyclist who speeds up from 18 km/h to 54 km/h in 5.0 s goes from 5.0 m/s to 15 m/s, so a = (15 − 5.0)/5.0 = 2.0 m/s². Subtracting the km/h values and dividing by the seconds gives 7.2, which is in km/h per second, written as if it were m/s². Mass goes into F = ma in kilograms, so 500 g is 0.500 kg.

Units matter outside homework too. NASA's investigation board traced the 1999 loss of the Mars Climate Orbiter to ground software that didn't translate English units into metric units (NASA, 1999).

Vectors

To split a vector into components, measure the angle from the axis you're working along. A plane flying at 200 km/h, 30° north of east, has an east component of 200 cos 30°, about 173 km/h, and a north component of 200 sin 30° = 100 km/h. If the angle is measured from north instead, sine and cosine trade places.

Vectors also don't add like plain numbers. Forces of 3 N east and 4 N north combine to 5 N, about 53° north of east, because √(3² + 4²) = 5. Adding them to get 7 N treats the two directions as one.

How Ally helps as you work

In Homework help, when you lift your pencil for about two seconds, or tap Check my work, Ally reads your newest line with the lines above it. If everything so far follows, Ally stays quiet. When a step doesn't follow, Ally highlights the term that went wrong and asks about it in one short sentence. The hint is meant to point at the slip and leave the corrected equation, the number and the final answer to you.

Take the tension problem above. If your line reads 30 − 9.8 = 2.0a, Ally highlights the 9.8 and asks a question like “How does the box's weight depend on its mass?” You see that the weight is mg, write 19.6 in its place and carry on.

Or say you got a = 6.25 m/s² for the braking car, and you hold the talk button to ask whether it should be negative. Ally asks a question like “When the car slows down, which way does its acceleration point compared with its velocity?” and lets you decide the sign.

A few more things that help in physics:

• Units are part of the answer. Ally treats a physics answer as finished only when it has a number and its units, such as 5.2 m/s².
• Ask why out loud. If you ask why the normal force on a ramp isn't just mg, Ally explains that it balances only the part of the weight pressing into the ramp, and asks which part of mg that is.
• A sketch when you can't picture it. You can ask Ally to draw the setup, such as the forces on a block on a ramp or a velocity split into components.
• Memory of past slips. If you've written g for the weight before, Ally can remind you what helped last time.

Common physics mistakes, one page each

• Unit conversions: setting up conversion factors so the units cancel
• The sign of gravity: why a = −9.8 m/s² when up is positive
• Vector components: when to use sine and when to use cosine

Comparing alai with other apps

For other ways to get help, see alai vs ChatGPT study mode and alai vs Khanmigo. For what the research says about hints that stop short of the answer, see homework help without answers.