Connect projectile components through one clock

1. Orient: Connect projectile components through one clock

With uniform gravity and negligible air resistance, horizontal acceleration is zero while vertical acceleration is −g for an upward-positive axis. This separates the component equations: x=vₓ₀t and y=vᵧ₀t−0.5gt² for an origin at launch. The components share one elapsed time; they are not two separate trips.

2. Explain the model

At the apex the vertical velocity vᵧ=vᵧ₀−gt is zero, so t_apex=vᵧ₀/g. Horizontal velocity does not become zero there. Substituting the apex time into y gives maximum height vᵧ₀²/(2g). When launch and landing heights agree, solving y=0 gives t=0 and the later root T=2vᵧ₀/g.

The equal-height range is vₓ₀T. Changing only horizontal launch velocity therefore changes range but not flight time or maximum height. Changing launch speed while keeping its angle fixed changes both components, so it is a different intervention. For an elevated landing platform, solve the vertical equation for that height instead of reusing the equal-height formula.

The model uses g=10 m/s² as an assigned rounded teaching value. It omits drag, spin, changing gravity and object size. It is not a safe launch-design tool. Work with the simulation or teacher-supplied footage; no projectile construction or physical launch is required. Pure vertical free fall is the same vertical equation with zero horizontal velocity. For a drop from rest through 20 m using downward-positive coordinates and g=10 m/s², 20=5t² gives t=2 s and downward speed 20 m/s. With an upward-positive axis, displacement and acceleration are both negative and give the same positive elapsed time.

3. Work an example

For vₓ₀=6 m/s and vᵧ₀=8 m/s, T=1.6 s, range=9.6 m, apex time=0.8 s and height=3.2 m. At the apex velocity is (6,0) m/s, not zero. A second launch with horizontal component 3 m/s has half the range but the same vertical history.

4. Predict and investigate

Predict first: If horizontal velocity doubles while upward velocity is fixed, which outputs double and which remain unchanged?

Hold upward velocity at 8 m/s and compare horizontal components 2,4 and 8 m/s. Then hold the horizontal component fixed and vary the vertical component. Use ratios to test the range, flight-time and height equations. State the condition needed for the return-time formula. Use the lesson experiment controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

Using g=10 m/s², a projectile has horizontal launch velocity 7 m/s and vertical launch velocity 5 m/s. It lands at launch height. Find its horizontal range.

Your numerical result (m) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

How would you change the solution if the projectile landed 1 m above its launch height? Which component determines candidate landing times?

Record a claim, numerical or graphical evidence, assumptions, a limitation and a changed-case prediction. A supplied synthetic investigation is not evidence that you performed a physical laboratory.

Framework and source notes

Original teaching prose, cases and synthetic datasets. Framework mapping: 1.5.A; 1.5.B. College Board AP Physics 1 CED, Unit 1, checked 30 October 2026. This mapping is a content cross-reference, not AP authorization or a claim of full course completion.