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Learn through experiments

Hit the target without hitting the ceiling

Can two trajectories reach similar distances but have different feasibility?

Entering midway? Revisit Why double speed needs more than double space for the preceding worked example, or start here and use the explanation below.

1 · Try

Start with a concrete question

Original synthetic scenario. A virtual projectile launches and lands at the same height. Gravity is set to 10 m/s² to make arithmetic inspectable.

Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.

Go directly to the experiment · Review basic mathematics

2 · Predict

What do you expect?

At 10 m/s, compare 30° and 60°: will range and maximum height both match?

3 · Experiment

Change one thing. Inspect what happens.

  1. Compare 30° and 60° at the same speed; inspect horizontal and vertical components.
  2. Set the ceiling to 2 m and choose a trajectory that satisfies it.
  3. Adjust speed to reduce target error while keeping the ceiling constraint.
  4. Explain why the smallest error among all trials may be an unacceptable design.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Range10 m
Maximum height2.5 m
Target absolute error0 m
Ceiling exceeded0 m

Compare the current results

Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.
010
  • Range (m)10
    Range (m): 10Horizontal bar from zero to 10, on the shared zero to 10 numeric axis.
  • Target distance (m)10
    Target distance (m): 10Horizontal bar from zero to 10, on the shared zero to 10 numeric axis.
  • Maximum height (m)2.5
    Maximum height (m): 2.5Horizontal bar from zero to 2.5, on the shared zero to 10 numeric axis.
  • Ceiling (m)4
    Ceiling (m): 4Horizontal bar from zero to 4, on the shared zero to 10 numeric axis.

Follow the calculation

  1. Horizontal speed = 10 cos(45°) = 7.071 m/s; initial vertical speed = 7.071 m/s.
  2. Using toy gravity 10 m/s², flight time = 2 × 7.071 / 10 = 1.414 s.
  3. Range = 7.071 × 1.414 = 10.000 m.
  4. Peak height = 7.071² / 20 = 2.500 m.
  5. Constraint check: peak 2.500 ≤ ceiling 4 m.

The ceiling constraint passes; inspect target error separately. A feasible arc need not hit the target.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

What the experiment explains

Original synthetic scenario. A virtual projectile launches and lands at the same height. Gravity is set to 10 m/s² to make arithmetic inspectable.

Resolve launch speed into horizontal and vertical components. Horizontal speed remains constant; gravity reduces vertical speed until the peak and then accelerates the fall.

At 10 m/s and 45°, both components are about 7.071 m/s. Flight time is 1.414 s, range is 10 m, and peak height is 2.5 m.

Complementary angles have the same ideal range but different peak heights. Constraints change which solutions are acceptable before optimization.

Read the relationship

vx = v cos θ; vy = v sin θ; flight time = 2vy/g; range = vx × time; peak height = vy²/(2g). θ is degrees converted to radians; g = 10 m/s².

Inspect the evidence

The numerical trace is recalculated from your current settings. Compare a controlled pair of trials before changing several inputs together. Inspect intermediate quantities as well as final metrics; state which assumptions make the comparison meaningful.

Boundaries of the model

  • No drag, spin, obstacles other than a flat ceiling, or unequal landing height.
  • Virtual design practice, not a physical launch recommendation.

Return to the physics and engineering course and project to connect this mechanism to the broader investigation.

5 · Apply

Try a fresh case

A fresh launch has horizontal speed 6 m/s and vertical speed 8 m/s, with gravity 10 m/s². What is its range at its starting height?

This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.

Take the investigation further

Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.

School investigation

Draw the horizontal and vertical motion separately and annotate units.

College investigation

Add numerical drag and validate the zero-drag limit against the closed-form result.

Workplace investigation

Write a constrained-design brief with model validity, tolerances, and a verification plan.

6 · Continue

Keep the evidence, choose your next step

Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.