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Impulse & Momentum Calculator

Apply the impulse–momentum theorem: J = FΔt = Δp to find velocity changes and momentum

Parameters

kgⓘ
m/sⓘ
Nⓘ
sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Impulse:
10.00;N⋅s10.00;N·s
Momentum Change:
10.00;kg⋅m/s10.00;kg·m/s
Final Velocity:
8.00;m/s8.00;m/s
Initial Momentum:
6.00;kg⋅m/s6.00;kg·m/s
Final Momentum:
16.00;kg⋅m/s16.00;kg·m/s

Examples

Hockey puck

2 kg puck at 3 m/s hit by 20 N for 0.5 s.

  • Impulse: 10.0010.00
  • Final Velocity: 8.008.00

Visualization

Impulse and the Momentum Theorem

Linear momentum p = mv measures how difficult it is to stop a moving object. It is a vector: direction matches velocity. Units are kg·m/s, equivalent to N·s.

The impulse–momentum theorem states that impulse J equals the change in momentum: J = Δp = m(v − u). Impulse is the cumulative effect of force over time.

For constant force F during contact time Δt: J = FΔt. A large force over a short time (bat hitting a ball) can equal a smaller force over longer time (same Δp).

Graphically, impulse is the area under the force–time curve. Variable forces require integration; constant F is the common textbook case.

When the net external force on a system is zero, total momentum is conserved—essential for collisions and explosions (analyzed in other calculators).

Key Concepts

  • p = mv — linear momentum
  • J = FΔt = Δp — impulse–momentum theorem
  • Δp = m(v − u) — momentum change
  • 1 N·s = 1 kg·m/s — unit equivalence
  • Vector nature: direction matters for 2D/3D
  • Conservation when F_ext = 0

Real-World Applications

  • Sports: bats, rackets, and ball impacts
  • Airbags and crumple zones (longer Δt, smaller peak F)
  • Rocket thrust: exhaust momentum per second
  • Collision analysis and safety engineering
  • JEE/NEET problems on force–time graphs

Explore Further

More mechanics tools

Physics Equations

Impulse:
J=FΔtJ = F \Delta t
Momentum Change:
Δp=m(v−u)\Delta p = m(v - u)
Final Velocity:
v=u+FΔtmv = u + \frac{F \Delta t}{m}

Step-by-Step Solution

See how the main results are calculated.

1

Calculate Initial Momentum

Linear momentum is mass times velocity:

Equation:

pi=mv0p_i = mv_0

Calculation:

pi=(2)(3)=6.00 kg\cdotpm/sp_i = (2)(3) = 6.00 \text{ kg·m/s}

Result:

pi=6.00 kg\cdotpm/sp_i = 6.00 \text{ kg·m/s}

Explanation:

Momentum is a vector. Use consistent sign convention for direction (e.g. right = positive).

2

Calculate Impulse

For a constant force acting over contact time Δt:

Equation:

J=FΔtJ = F \Delta t

Calculation:

J=(20)(0.5)=10.00 N\cdotpsJ = (20)(0.5) = 10.00 \text{ N·s}

Result:

J=10.00 N\cdotpsJ = 10.00 \text{ N·s}

Explanation:

Impulse equals the area under the force–time graph. 1 N·s = 1 kg·m/s, the same unit as momentum change.

3

Apply Impulse–Momentum Theorem

Impulse equals change in momentum:

Equation:

J=Δp=m(v−u)J = \Delta p = m(v - u)

Calculation:

Δp=10.00 kg\cdotpm/s\Delta p = 10.00 \text{ kg·m/s}
m(v−u)=2(v−3)⇒v−u=5.00 m/sm(v - u) = 2(v - 3) \Rightarrow v - u = 5.00 \text{ m/s}

Result:

Δp=10.00 kg\cdotpm/s\Delta p = 10.00 \text{ kg·m/s}

Explanation:

The theorem is Newton’s second law integrated over time. A large force for a short time can produce the same Δp as a smaller force for longer.

4

Find Final Velocity

Rearrange for final speed:

Equation:

v=u+Jmv = u + \frac{J}{m}

Calculation:

v=3+10.002=8.00 m/sv = 3 + \frac{10.00}{2} = 8.00 \text{ m/s}

Result:

v=8.00 m/sv = 8.00 \text{ m/s}

Explanation:

Final momentum p_f = 16.00 kg·m/s. Check: p_f − p_i = 10.00 = J.

Frequently Asked Questions (FAQ)

What are the units of impulse?

N·s, which is dimensionally the same as kg·m/s (momentum).

Is impulse a vector?

Yes—impulse has the same direction as the constant force (or is the integral of the force vector for variable F).

How do airbags reduce injury?

They increase contact time Δt, so for the same Δp the average force F = J/Δt is smaller.

Can impulse change momentum without changing speed?

Yes, if only the direction of velocity changes (vector Δp).

Practice MCQs

  1. Doubling contact time with same force:
  2. Doubling mass with same impulse:
  3. Impulse equals:

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