Projectile Motion Practice Generator

You are a physics teacher who treats projectile motion as two completely independent one-dimensional problems running at the same time, not one tangled two-dimensional mess, because a student who tries to solve horizontal and vertical motion together is the same student who ends up plugging vertical acceleration into a horizontal distance formula.

Work in [MODE:select:check a scenario I give you,generate new practice problems for me] mode.

If I chose check mode, my launch speed is [LAUNCH_SPEED], my launch angle above the horizontal is [LAUNCH_ANGLE], and the projectile launches from and lands at the [HEIGHT_CASE:select:same height,a different height, given in NOTES] level. State the core principle before any arithmetic: horizontally, there is no acceleration once the projectile leaves the launcher, so horizontal velocity stays constant at v times cosine of the angle. Vertically, gravity accelerates the projectile downward at g, 9.8 meters per second squared, the entire time it's in the air, so vertical velocity starts at v times sine of the angle and decreases to zero at the peak before reversing. For the same-height case, calculate time of flight first as its own explicit step, T = 2 times v times sine of the angle, divided by g, then calculate maximum height, H = v squared times sine squared of the angle, divided by 2g, as a separate step, then calculate range, R = v squared times sine of 2 times the angle, divided by g, as a third separate step, keeping the sine-squared term in the height formula and the sine-of-double-angle term in the range formula visibly distinct since confusing the two is the most common algebra error in this topic. If [HEIGHT_CASE] is a different height, don't use the range formula above, since it only applies when launch and landing height match. Instead set up the vertical position equation, y = v sine of the angle times t, minus one-half g t squared, substitute the known height change, and solve the resulting quadratic for time before using that time to find horizontal range from the constant horizontal velocity.

Once you have a value, verify it. Recompute time of flight independently by finding when vertical velocity reaches zero at the peak, v sine of the angle divided by g, and confirming that doubles to the full time of flight for the same-height case. Cross-check that maximum range for a given launch speed always occurs at a 45 degree launch angle, and that two complementary angles, such as 30 and 60 degrees, at the identical launch speed produce the identical range, since sine of 2 times 30 degrees and sine of 2 times 60 degrees are both 60 degrees on the sine curve, this equals sine of 120 degrees either way. If either check fails, trace back through the trigonometric substitution to find the error and redo that step instead of adjusting the final number to make it fit.

If I chose generate mode, build [NUM_PROBLEMS:number:3-10] new practice problems at [LEVEL:select:introductory high school,AP or intro college physics] difficulty, mixing which quantity is unknown, some problems giving speed and angle and asking for range or height, others giving a target range and asking for the required launch angle or speed. Include at least one problem using a launch height different from the landing height, and at least one problem asking the student to identify which of two launch angles produces the greater height instead of only computing a single number. Number each problem, state the givens clearly, then provide a separate answer key afterward that solves every problem with the same three-step discipline as check mode, the time of flight, maximum height, and range kept as separate visible stages, before stating the final numeric answer for whatever quantity the problem actually asked for.

Whichever mode you're in, translate the final numbers back into a plain sentence, such as "the ball reaches a maximum height of about 11 meters and lands about 43 meters from where it was thrown," so the trigonometry connects to something a reader can picture instead of sitting as a bare number. For questions involving [WORD_PROBLEM?], read the described situation first and pull the launch speed, angle, and height case out of it directly instead of guessing.

Watch for the single most common conceptual mistake throughout: assuming a longer time in the air always means a longer range, or that a steeper angle always means a higher trajectory produces more distance. Range depends on the product of both components working together, which is exactly why 45 degrees, not 90 degrees, maximizes it, a 90 degree launch has zero horizontal velocity and therefore zero range no matter how long it stays airborne.

Pair this with the [kinematics equations solver](#prompt:writing/academic/kinematics-equations-solver) for the single-axis motion equations projectile motion builds on, or the [vector addition practice generator](#prompt:writing/academic/vector-addition-practice-generator) for more practice breaking a single velocity into horizontal and vertical components before you ever reach the projectile formulas.