Archery Arrow Momentum Calculator
Created by: Ethan Brooks
Last updated:
Calculate arrow momentum and kinetic energy from complete-arrow mass and measured speed, with an optional second-setup comparison.
Archery Arrow Momentum Calculator
ArcheryMeasured mass × measured speed
What does this calculator measure?
An arrow momentum calculator multiplies the complete arrow's mass by its measured speed to find linear momentum magnitude. Enter grains and feet per second or grams and meters per second. The calculator converts those measurements to a consistent SI basis and reports momentum in kilogram-meters per second. It also shows translational kinetic energy in joules, which uses the same mass but the square of speed rather than speed alone.
Use the mass of the complete arrow at the time of the speed measurement. A bare-shaft specification excludes components, so it is not interchangeable with a finished-arrow scale reading. Likewise, an advertised bow speed describes a particular rating setup rather than your measured arrow. This tool deliberately asks for the speed you measured and does not calculate a new speed when you change the arrow's mass.
The optional comparison treats two setups as separate sets of observations. Each needs its own complete-arrow mass and speed. Results show both momenta and their signed difference, while a chart makes the momentum comparison easy to see. A positive B-minus-A difference means the second entered setup has more momentum. It does not mean that setup is more accurate, more efficient, or more appropriate for a particular use.
Keep the measurement location and procedure with your equipment notes. A speed recorded near the bow and one recorded farther along the flight path are different observations, even for the same arrow. This calculation does not add air resistance, estimate retained speed or model a collision. It reports physical quantities associated with the measurements supplied, without turning momentum or energy into a penetration prediction, impact force or equipment recommendation.
How the calculation works
Convert grains to grams with 0.06479891 grams per grain, then divide grams by 1,000 for kilograms. Convert feet per second to meters per second by multiplying by 0.3048. Linear momentum magnitude is kilograms multiplied by meters per second. Translational kinetic energy is one half of the mass in kilograms multiplied by the square of speed in meters per second. Calculations use the entered precision before the displayed values are rounded.
For two setups, subtract A's momentum from B's momentum. The relative change is that difference divided by A's momentum, multiplied by 100. If A has zero momentum, a relative percentage is undefined; the absolute difference remains meaningful. Zero speed is supported as an at-rest example, but mass must be positive. The model uses speed, a nonnegative magnitude, and therefore does not assign an arrow's direction or calculate vector components.
Formula and symbols
p = m × v; KE = m × v² / 2; difference = pB − pA; relative change = 100 × difference / pA when pA > 0.
- m: Complete-arrow mass (kg internally)
- v: Measured speed at the recorded position (m/s internally)
- p, KE: Momentum magnitude and translational kinetic energy (kg·m/s and J)
How to use the calculator
- Choose units. Select grains and feet per second or grams and meters per second. Existing measurements convert when the selection changes.
- Enter measured mass and speed. Use the complete-arrow mass and speed measured for that same setup. Retain the measurement location in your notes.
- Optionally compare another setup. Enable comparison and supply a separate complete mass and speed pair for setup B.
- Review the two quantities. Calculate momentum and kinetic energy, retaining their distinct units and the assumptions beside the result.
Worked examples
A direct SI calculation
A complete arrow has a measured mass of 20 grams and speed of 50 meters per second. Convert 20 grams to 0.020 kilograms. Momentum is 0.020 × 50 = 1 kilogram-meter per second. Kinetic energy is 0.5 × 0.020 × 50² = 25 joules. The two values describe the same observation but have different dimensions. Neither gives an impact force, since no stopping time, stopping distance or collision model has been entered.
Comparing two measured setups
Use the first example for A, then enter a separately measured setup B of 25 grams at 40 meters per second. B's momentum is also 1 kilogram-meter per second, so the momentum difference is zero. Its kinetic energy is 20 joules rather than A's 25 joules. This example shows why equal momentum does not imply equal kinetic energy. The two speeds are observations supplied for the example; the calculator has not predicted the second speed from its heavier mass.
An at-rest reference
For 20 grams at zero meters per second, both momentum and kinetic energy are zero. Comparing that result with the first moving-arrow example produces an absolute difference of 1 kilogram-meter per second. Percentage change is left undefined because its denominator would be zero. This mathematical boundary is useful for checking the calculation, not as a substitute for a measured moving-arrow speed.
Practical applications
- Record mass and speed measurements together. The resulting momentum and energy provide two reproducible quantities for an equipment notebook, while the table preserves the inputs that produced them. Add your own measurement location and setup identifiers to that record.
- Compare independently measured setups without inventing a mass-to-speed relationship. Supply a complete pair of readings for each setup, then inspect the absolute momentum difference. Keep the test procedure comparable if you want the difference to answer a useful question.
- Translate grain-based and metric measurements into one calculation. Unit switching converts existing input values, allowing a scale reading and speed reading to be expressed consistently. Both unit modes produce the same physical momentum and energy for the same observation.
- Understand the difference between momentum and energy. Change one input at a time as a clearly labeled mathematical scenario, and note that momentum depends linearly on speed while energy depends on speed squared. A scenario is not a measured performance prediction.
- Check an arithmetic result from a notebook or spreadsheet. Use the worked SI example as a reference, then verify that your other calculation converts grams to kilograms and feet per second to meters per second before using the equations.
- Keep a comparison traceable when sharing results. The exported result includes the entered masses, speeds, equations and scope. That record helps another reader distinguish measurements from assumptions and prevents an isolated momentum number from losing its original context.
Tips for consistent measurements
Verify the scale's unit indicator and use the assembled arrow's mass. Take the speed from a suitable measurement record, and keep its measurement position consistent across comparisons. Avoid rounding intermediate values in a separate worksheet. A reading to the nearest whole grain does not become more physically precise just because the calculator can display extra decimal places after converting it to grams.
When comparing A and B, enter both values for B before calculating. Turning comparison off ignores its hidden fields. If you change an input or unit, the previous result clears so it cannot be mistaken for the updated scenario. Use the signed difference to describe the comparison; do not convert a higher value into an unsupported claim about accuracy, impact force or target performance.
Frequently asked questions
What units does arrow momentum use?
The result is in kilogram-meters per second, written kg·m/s. This is equivalent to newton-seconds, N·s, but it is not a force in newtons. Inputs can be grains and feet per second or grams and meters per second. The calculator converts them to kilograms and meters per second before applying the momentum equation.
Is momentum the same as kinetic energy?
No. Momentum magnitude is mass multiplied by speed, while kinetic energy is half the mass multiplied by speed squared. The calculator reports them separately with different units. Two arrows can have equal entered momentum and different kinetic energy, as the comparison example shows. Neither quantity alone specifies the force during a collision or its outcome.
Can I use my bow's advertised speed?
Use a measured speed for the actual arrow and setup you are recording. An advertised rating may refer to different test conditions and is not automatically a measurement of your arrow. If you enter a hypothetical speed for exploration, treat the result as that hypothetical scenario. The tool does not correct a rating to match your equipment.
Does this predict momentum farther downrange?
No. The output corresponds to the speed you enter at its measurement location. The calculation does not contain an aerodynamic drag model, a distance input or a velocity-loss estimate. To record another location, use a corresponding measured speed and retain that location with your notes. Do not relabel a near-bow result as a downrange measurement.
Why is percentage change missing when setup A is at rest?
A relative change divides the momentum difference by setup A's momentum. If A has zero speed, its momentum is zero, so that division is undefined. The calculator still shows the two momenta and their absolute difference. Enter a nonzero measured baseline if a relative comparison is appropriate; no arbitrary replacement denominator is inserted.
Can a higher momentum result tell me which arrow to choose?
This tool makes no equipment selection. It compares the supplied mass and speed observations, while shaft compatibility, measured balance and other equipment requirements remain separate. A higher momentum value does not establish accuracy, suitability or a particular impact result. Use the calculation as a documented measurement alongside the relevant manufacturer's guidance and your actual setup records.
Sources and measurement scope
- OpenStax: College Physics 8.1 — Linear Momentum and Force — Online College Physics; Linear momentum, p = mv. Accessed 2026-10-06. Classical momentum magnitude for measured mass and speed; no impact or penetration prediction.
- OpenStax: College Physics 7.2 — Kinetic Energy — Online College Physics; Kinetic energy, KE = mv²/2. Accessed 2026-10-06. Translational kinetic energy, not stored bow energy or a target outcome.
- NIST: SP 811 Appendix B.9 — mass, force and velocity — SP 811 online Appendix B.9; Force (footnote 23); Mass and Moment of Inertia; Velocity. Accessed 2026-10-06. Grains, feet per second and pounds-force converted to SI. Exact lbf derived from 0.45359237 kg and standard gravity 9.80665 m/s².
Worked examples use illustrative measurements. These tools do not select equipment or prescribe a preferred setup. Any optional manufacturer comparison uses the reference entered by the user.