Momentum Calculator

Solve for linear momentum, mass, or velocity using \(p = m v\). Includes common SI and imperial units, quick stats, and steps.

Configuration

Choose which variable you want to compute and the units for the main result.

Inputs

Result

Practical Guide

Momentum Calculator

Use this Momentum Calculator to solve for linear momentum, mass, or velocity with clean unit handling. This guide explains the core physics, shows when to use different momentum methods, and helps you sanity-check results for real engineering and classroom problems.

6–8 min read Updated 2025

Quick Start

  1. 1 Choose what you want to solve for: momentum \(p\), mass \(m\), or velocity \(v\).
  2. 2 Enter the two “known” variables the equation needs. The solved-for row is hidden automatically.
  3. 3 Pick units next to each input (kg, lbm, slug for mass; m/s, ft/s, mph, km/h for velocity).
  4. 4 If solving for momentum, select your preferred result units (kg·m/s, N·s, lbf·s, or slug·ft/s).
  5. 5 Confirm direction: use a negative velocity if the object moves opposite your chosen positive axis.
  6. 6 Review the result row and Quick Stats. Quick Stats show the SI-normalized values and kinetic energy.
  7. 7 Open “Show Steps” to see the exact rearranged equation and substituted numbers.

Tip: Treat momentum as a vector. If you only care about magnitude, enter positive values; if direction matters, keep signs consistent.

Common mistake: Mixing mass units (lbm vs slug) or velocity units (mph vs ft/s) without converting. The calculator handles this if you pick the right unit selectors.

The calculator is based on linear momentum: \[ p = m v \] where \(p\) is momentum, \(m\) is mass, and \(v\) is velocity along a chosen axis. Internally, the tool converts everything to SI (kg, m/s) before solving.

Choosing Your Method

“Momentum” problems show up in mechanics, vehicle safety, robotics, ballistics, manufacturing impacts, and any scenario involving motion and forces over time. The calculator uses the simplest form \(p = m v\), but engineers often extend this idea in a few standard ways. Here’s how to decide which approach matches your problem.

Method A — Linear Momentum (This Calculator)

Use \(p = m v\) when you know mass and velocity at a specific instant, or when the system behaves like a single lumped mass.

  • Fast and direct; ideal for “at this speed, what is the momentum?” questions.
  • Works for translation in 1D or along a chosen axis in 2D/3D.
  • Great for sizing buffers, estimating impact severity, or comparing moving objects.
  • Doesn’t include time or force directly.
  • Requires a clear sign convention for direction.
\(p = m v\)

Method B — Impulse–Momentum

Use this when force acts over a time interval and changes velocity, such as collisions, braking, or press forming.

  • Connects force and time to the change in momentum.
  • Handles non-constant forces via average force or force-time curves.
  • Most common for impact and crash-analysis problems.
  • Requires time history or average force data.
  • Not what this calculator is solving directly (but its momentum output plugs in).
\[ J = \int F\,dt = \Delta p \]

Method C — Conservation of Momentum (Systems)

Use this when internal forces dominate and external impulses are negligible (e.g., explosions, docking, or two-body collisions).

  • Powerful for multi-object problems.
  • Works even when forces are unknown during collision.
  • Standard check in crash/impact and recoil scenarios.
  • Needs a clear system boundary and external impulse estimate.
  • Must track momentum vectors for each object.
\[ \sum p_{\text{before}} = \sum p_{\text{after}} \]

If your question is “what is the momentum of this object right now?”, Method A is the right tool—exactly what this calculator provides. If you’re analyzing a collision or braking event, compute momenta for each phase with this calculator, then use Methods B or C to finish the analysis.

What Moves the Number the Most

Momentum is simple algebraically, but in practice it is very sensitive to how you measure or assume inputs. These are the dominant levers:

Velocity magnitude

Momentum scales linearly with \(v\). Doubling velocity doubles momentum. In impact problems, this often dominates uncertainty.

Mass definition

Use the correct mass for the moving body. For vehicles, include payload; for rotating assemblies translating, include effective moving mass only.

Direction / sign convention

Momentum is a vector. A negative sign means momentum is opposite your chosen positive axis and affects conservation sums.

Imperial unit choice

lbm (mass) is not the same as lbf (force). If you’re in imperial, use lbm or slugs consistently. This calculator converts for you when selected correctly.

Component vs total velocity

For angled motion, use axis components: \(p_x = m v_x\), \(p_y = m v_y\). Don’t mix magnitudes with components.

Time window (for impulse problems)

If you later compute \(\Delta p\), the chosen “before” and “after” velocities strongly set the impulse estimate.

Engineering intuition: If two objects have the same speed, the heavier one has more momentum. If two objects have the same mass, the faster one has more momentum.

Worked Examples

Example 1 — Solve for Momentum

  • Object: Cart on a test track
  • Mass: \(m = 12\ \text{kg}\)
  • Velocity: \(v = 3.5\ \text{m/s}\) (positive direction)
  • Solve For: Momentum \(p\)
1
Convert to SI (already SI): \(m=12\ \text{kg}\), \(v=3.5\ \text{m/s}\).
2
Apply core equation: \[ p = m v \]
3
Substitute values: \[ p = (12)(3.5) = 42\ \text{kg·m/s} \]
4
If you select N·s as output, result is the same numerically: \(42\ \text{N·s}\).

Answer: \(p = 42\ \text{kg·m/s}\). Quick Stats will also show kinetic energy: \[ E_k = \tfrac12 m v^2 = \tfrac12(12)(3.5^2) \approx 73.5\ \text{J} \]

Example 2 — Solve for Mass

  • Measured momentum: \(p = 180\ \text{N·s}\)
  • Velocity: \(v = 25\ \text{m/s}\)
  • Solve For: Mass \(m\)
1
Convert to SI: \(p=180\ \text{kg·m/s}\), \(v=25\ \text{m/s}\).
2
Rearrange: \[ m = \frac{p}{v} \]
3
Substitute: \[ m = \frac{180}{25} = 7.2\ \text{kg} \]
4
If you change mass units to lbm, the calculator converts: \[ m \approx \frac{7.2}{0.4536} \approx 15.9\ \text{lbm} \]

Answer: \(m = 7.2\ \text{kg}\) (about \(15.9\ \text{lbm}\)).

Example 3 — Solve for Velocity with Direction

  • Momentum: \(p = -60\ \text{kg·m/s}\)
  • Mass: \(m = 5\ \text{kg}\)
  • Solve For: Velocity \(v\)
1
Convert to SI (already SI): \(p=-60\), \(m=5\).
2
Rearrange: \[ v = \frac{p}{m} \]
3
Substitute: \[ v = \frac{-60}{5} = -12\ \text{m/s} \]

Answer: \(v = -12\ \text{m/s}\). The negative sign means motion is opposite your chosen positive axis.

Common Layouts & Variations

Real problems vary by application. The table below shows common configurations and how the same momentum relationship is used.

Application / ConfigurationGiven InputsTypical Use CaseNotes & Pros/Cons
Single translating mass (1D)\(m, v\)Conveyors, carts, projectiles along a lineMost direct case. Keep sign convention consistent.
Two-body collision (system)\(m_1, v_1, m_2, v_2\)Crash tests, bumpers, dockingCompute each \(p_i = m_i v_i\), then apply conservation.
Impulse event\(F(t), \Delta t\) or \(F_{\text{avg}}, \Delta t\)Braking, stamping, hammer blowsUse \(\Delta p = J\). This calculator gives “before/after” momentum.
Angled / 2D motion\(m, v_x, v_y\)Robotics, drones, fluid jetsCompute components \(p_x=m v_x\), \(p_y=m v_y\). Resultant \(p=\sqrt{p_x^2+p_y^2}\).
High-speed rotating part translatingEffective \(m\), axial \(v\)Spindles, flywheels on linear stagesUse only translating mass for linear momentum; treat rotation separately via angular momentum.
Imperial engineering workflowlbm or slug, ft/s or mphUS vehicle and structural impact workPick correct mass units. lbf·s is impulse; calculator converts to SI internally.

If your scenario includes deformation, rebound, or multiple phases, compute momentum at each phase and compare changes—momentum itself is still \(m v\), but the engineering insight comes from how it changes.

Specs, Logistics & Sanity Checks

Momentum calculations are often used to size components (buffers, catch systems, brakes) or to check safety margins. Before you trust a result, run these checks.

Input Verification

  • Mass matches what is actually moving (include payload, exclude fixed supports).
  • Velocity is at the correct instant (pre-impact vs post-impact).
  • Axis and sign are consistent with your free-body diagram.
  • Units chosen in each selector match the measurement source.

Engineering Reality Checks

  • Compare to a baseline: doubling speed should double momentum.
  • Heavier object at same speed should yield proportionally higher \(p\).
  • Momentum values for vehicles should align with published crash data ranges.
  • If solving for mass or velocity, confirm the answer is physically plausible.

Design / Field Notes

  • For impacts, use momentum with a stopping distance/time model to size loads.
  • For safety barriers, compute worst-case momentum (max mass × max speed).
  • For manufacturing drops, include vertical velocity at contact (from height).
  • Document assumptions: friction losses, external impulses, or neglected masses.

Limitation: This calculator assumes classical (non-relativistic) speeds. At a significant fraction of the speed of light, use relativistic momentum \(p=\gamma m v\).

Good practice: If you are using momentum to size a safety system, add a conservative factor (e.g., 1.2–1.5×) to cover measurement uncertainty.

Frequently Asked Questions

What is the momentum equation used in this calculator?
The calculator uses linear momentum: \[ p = m v \] It can rearrange to \(m=p/v\) or \(v=p/m\) depending on what you select.
Is momentum the same as impulse?
Not exactly. Impulse \(J\) is the change in momentum: \[ J = \Delta p \] In SI, both share units of N·s (or kg·m/s), so they’re numerically comparable.
Why does the calculator show lbf·s instead of lbm·ft/s?
In imperial engineering, momentum is often reported as impulse in lbf·s. The calculator converts internally to SI, so you can safely use lbf·s or slug·ft/s as output units.
How do I handle direction or negative momentum?
Choose a positive axis. If the object moves opposite that axis, enter a negative velocity (or momentum). The sign will carry through your result and any conservation sums.
Can I use this for 2D or angled motion?
Yes—compute components separately. For example, enter \(v_x\) to get \(p_x=m v_x\), then repeat for \(v_y\). Combine with \[ p=\sqrt{p_x^2+p_y^2} \] if you need magnitude.
What units should I use for mass in US customary problems?
Use lbm if your mass comes from weight divided by \(g\) or from a spec sheet in pounds-mass. Use slugs for dynamics textbooks or when your workflow is already in slug-based units. Pick the correct selector and the calculator converts automatically.
When would I need relativistic momentum instead?
If speeds are a significant fraction of the speed of light (roughly \(v > 0.1c\)), classical momentum starts to under-predict. Use \[ p=\gamma m v,\quad \gamma=\frac{1}{\sqrt{1-v^2/c^2}} \] which is outside this calculator’s scope.
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