Hoyt RX-9 Ultra scenario lab

Ballistic Energy

A SvelteKit rebuild of the prior slider model, now anchored to your updated chrono: 278.8 fps with the same 416.7 gr arrow, using a 69 lb primary draw-weight anchor and a 67 lb bad-scale alternate.

Exact KE + momentum math Tunable FPS heuristics Arrow sweep + target curves

Measured arrow: 27.00 g

Anchor let-off 85%

Scenario now 69.0 lb · 27.5 in

The exact energy equations are fixed. The speed layer is a calibrated heuristic you can tune with draw-weight, draw-length, arrow-mass, and let-off sensitivities.

Primary estimate

282.8 fps

74.0 ft-lb · 0.523 slug-ft/s

67 lb alternate

286.8 fps

76.1 ft-lb · 0.530 slug-ft/s

Range band

282.8-286.8 fps

74.0-76.1 ft-lb if the bad-scale anchor is real

Scenario GPP

6.04 gpp

416.7 gr arrow at 69.0 lb

Required FPS

265.0 fps

Needed to hit 65 ft-lb with the current arrow

Required grains

325 gr

Needed to hit 65 ft-lb at 300 fps

Anchor

Measured setup

This is the chrono-verified baseline. The alternate draw weight captures the bad-scale uncertainty without changing the measured arrow or measured speed.

Anchor let-off

Scenario

What-if sweep

Change the current setup and compare the estimated speed, KE, momentum, and target thresholds against the measured anchor.

Scenario let-off

Heuristics

Tune the model

The physics stays fixed. Only the estimated FPS layer moves here. Lower let-off is treated as a positive speed bonus, with 85% as the baseline and 75% as the full bonus.

Estimated speed vs arrow weight

Scenario geometry: 69.0 lb, 27.5 in, 75%

69 lb anchor 67 lb alternate

Kinetic energy vs arrow weight

Because KE scales with velocity squared, small speed changes still matter.

69 lb anchor 67 lb alternate

Required FPS for the current KE target

Current target: 65.0 ft-lb. Compare your estimated FPS against the requirement at each arrow mass.

65 ft-lb target

Assumptions

What is fixed and what is fitted

Kinetic energy KE = grains × fps² / 450240
Momentum Momentum = grains × fps / 225400
Required FPS sqrt(target KE × 450240 / grains)
Required grains target KE × 450240 / fps²
  • The exact energy and momentum math is not approximate.
  • The speed model is a calibrated heuristic anchored to your measured 278.8 fps.
  • Let-off is modeled exactly like the corrected prior canvas: 85% baseline, 80% half bonus, 75% full bonus.
  • The 67 lb alternate does not change the chrono reading. It only changes the weight-origin assumption around that reading.

Current scenario delta from measured anchor: + 4.0 fps on the 69 lb anchor.

Field Manual

Help, equations, notation, and how to read the model

This section separates the parts that are exact physics from the parts that are fitted assumptions. The goal is to make every number on the page traceable and easy to reason about.

1. Exact ballistic equations

These are the non-negotiable parts of the model. If you know arrow mass and speed, the energy and momentum results come straight from the equations below.

Kinetic energy

KE = grains × fps2 450240

This is why speed matters so much: velocity is squared, so even small FPS gains can move KE more than people expect.

Momentum

p = grains × fps 225400

Momentum rises linearly with both mass and speed, so heavier arrows hold their ground here more than they do in pure speed comparisons.

Required speed for a target energy

fpsrequired = √ ( target KE × 450240 grains )

This tells you how fast a specific arrow must go to clear a chosen energy threshold.

Required arrow mass for a target energy at a chosen speed

grainsrequired = target KE × 450240 fps2

Useful when you want to say, “If I insist on 300 fps, how much arrow mass do I need at minimum to still reach 65.0 ft-lb?”

2. The fitted speed model

FPS is the heuristic part. Instead of pretending there is one perfect universal bow-speed equation, the app starts with your measured chrono and then moves away from it using tunable sensitivities.

fpsestimate = fpsmeasured + Δdraw weight × (fps per lb) + Δdraw length × (fps per inch) - Δarrow grains / 5 × (fps per 5 grains) + Δlet-off bonus + bias trim
fps measured

The baseline chrono reading. Right now that is your measured 278.8 fps.

Δdraw weight

The scenario draw weight minus the chosen anchor draw weight.

Δdraw length

The scenario draw length minus the measured draw length.

Δarrow grains

The scenario arrow mass minus the measured arrow mass. Heavier arrows subtract speed.

Δlet-off bonus

The difference between the scenario let-off bonus and the anchor let-off bonus.

bias trim

A small manual nudge for future chrono tuning, without changing the baseline measurement itself.

Let-off logic

The corrected canvas behavior is preserved: 85% is baseline, 80% gets half the configured bonus, and 75% gets the full configured bonus.

Bad-scale alternate

The 67 lb alternate does not overwrite your chrono result. It creates a second interpretation band around the same measured shot.

3. Worked example from the current sliders

Here is the model spelled out in plain English using the values currently on screen.

  1. Your measured anchor shot is 416.7 gr at 278.8 fps.
  2. The current scenario arrow is 416.7 gr, so the mass shift from measured is +0.0 gr.
  3. The current scenario draw weight shift from the primary anchor is +0.0 lb.
  4. The draw length shift from the measured setup is +0.0 in.
  5. After applying the current sensitivity settings, the model estimates 282.8 fps on the primary anchor and 286.8 fps on the 67 lb alternate.
  6. Using the exact KE equation, that becomes 74.0 ft-lb of kinetic energy and 0.523 momentum on the primary path.
  7. To hit your current target of 65.0 ft-lb with this arrow, the minimum speed needed is 265.0 fps.

If the estimated FPS sits above the required-FPS line, the setup clears the target. If it sits below it, you either need more speed, more draw force, or a different arrow mass strategy.

4. How to interpret the charts

Speed vs arrow weight

Read this as the simplest tradeoff curve in the app. As arrow mass rises, estimated speed falls. If you are chasing flatter trajectory or a round-number chrono result, this is the first chart to look at.

KE vs arrow weight

This shows where added mass starts to offset the speed loss. It is a cleaner chart for answering “Which arrow region gives me the energy floor I want without overshooting mass?”

Required FPS curve

This is the threshold chart. Your estimated speed markers should ideally land above the required line for the target energy you selected.

GPP

Grains per pound is simply arrow grains divided by draw weight. It is a useful sanity check for building a hunting arrow, but it does not replace chrono data or the energy equations.