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Impulse & Momentum Calculator

Impulse equals force times change in time

Solution

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Impulse (Force × Time)

Impulse measures the effect of a force applied over a period of time. The SI unit is the newton-second (N·s). Impulse equals the change in momentum of the object.

J = F · Δt

Momentum (Mass × Velocity)

Momentum is a property of a moving object equal to its mass times velocity. The SI unit is kg·m/s. Momentum is conserved in all collisions within a closed system.

p = m · v

Impulse from Mass and Velocity Change

An alternative form expressing impulse as mass times velocity change, useful when you know the mass and the change in speed rather than force and time.

J = m · Δv

Momentum Change from Force and Time

The impulse-momentum theorem states that the change in momentum equals the impulse applied. This connects force and time to the resulting momentum change.

Δp = F · Δt

How It Works

Impulse (J = F · Δt) measures the effect of a force applied over time, while momentum (p = m · v) measures an object’s motion. The impulse-momentum theorem connects them: the impulse on an object equals its change in momentum. This is why airbags work — they increase the collision time, reducing the peak force on the body.

Example Problem

A 0.145 kg baseball is pitched at 40 m/s and hit back at 50 m/s. The bat contacts the ball for 0.001 s. What average force does the bat exert?

  1. Δp = 0.145 × (50 − (−40)) = 0.145 × 90 = 13.05 N·s
  2. F = 13.05 / 0.001 = 13,050 N

When to Use Each Variable

  • Solve for Impulse (F x t) — when you know the force and duration of impact, e.g., calculating the impulse delivered by a rocket engine during a burn.
  • Solve for Momentum (m x v) — when you know the mass and velocity, e.g., finding the momentum of a moving vehicle for collision analysis.
  • Solve for Impulse (m x dv) — when you know the mass and velocity change, e.g., determining the impulse on a ball during a bat swing.
  • Solve for Momentum Change (F x dt) — when you know the applied force and time interval, e.g., finding how much momentum a braking force removes.

Key Concepts

The impulse-momentum theorem states that the net impulse on an object equals its change in momentum. This principle explains why extending the collision time (airbags, crumple zones) reduces peak force. Momentum is always conserved in closed systems, making it the key tool for analyzing collisions, explosions, and propulsion.

Applications

  • Automotive safety: designing airbags and crumple zones to reduce peak impact force
  • Sports science: analyzing bat-ball collisions to optimize equipment design
  • Aerospace: calculating rocket thrust and delta-v for spacecraft maneuvers
  • Ballistics: predicting projectile behavior and terminal impact forces

Common Mistakes

  • Forgetting to account for direction — velocity and momentum are vectors, so reversals double the change in momentum
  • Confusing impulse with force — impulse is force multiplied by time, not force alone
  • Assuming kinetic energy is conserved in all collisions — only momentum is always conserved; kinetic energy is conserved only in perfectly elastic collisions

Frequently Asked Questions

What is the difference between impulse and momentum?

Momentum is a property of a moving object (p = mv). Impulse is the change in momentum caused by a force over time (J = F·Δt). They have the same units (N·s or kg·m/s).

Why do airbags reduce injury in a crash?

Airbags increase the time over which momentum changes. Since J = F·Δt, a longer Δt means a smaller average force on the occupant for the same impulse.

Is momentum conserved in all collisions?

Yes, total momentum is always conserved in a closed system (no external forces). Kinetic energy, however, is only conserved in perfectly elastic collisions.

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

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