What is impulse in physics?

Impulse (J) is the product of a force and the time interval over which it acts — J = F·Δt. It equals the change in momentum of the object the force acts on, making it measured in N·s (equivalently kg·m/s).

What is the impulse-momentum theorem?

The impulse-momentum theorem states that the impulse delivered to an object equals its change in momentum — J = F·Δt = m·v_f − m·v_i. It is a direct consequence of Newton's second law integrated over time.

What units does impulse use?

The SI unit of impulse is the newton-second (N·s), which is identical to kg·m/s. Both express the same physical quantity; the choice is usually a matter of context.

How is impulse different from momentum?

Momentum (p = m·v) is a property of a moving object at an instant. Impulse is the cause that changes that momentum — it requires a force acting over a time interval. Numerically, impulse equals the change in momentum, not momentum itself.

Why is a large time interval important in safety engineering?

When a collision is unavoidable the impulse (change in momentum) is fixed by physics. Spreading the force over a longer Δt — as crumple zones, airbags and helmets do — reduces the peak force F = J/Δt, lowering injury risk. The calculator lets you model exactly this trade-off.

Can impulse be negative?

Yes. Impulse is a vector quantity. If the force opposes the direction of motion (e.g., braking), the impulse is negative, reducing the object's momentum. The calculator handles signed values correctly — just enter negative velocities or forces where appropriate.

Impulse Calculator — Gera Tools

The impulse calculator applies the impulse-momentum theorem — one of the most practically useful results in classical mechanics — to let you solve for any unknown in the relationship between force, time, mass, and velocity change. Whether you are designing a collision absorber, analysing a sports impact, or working through a kinematics exam question, this tool shows the full step-by-step working so you understand every number.

The physics behind the tool

Impulse (symbol J) is defined as the integral of a net force over the time interval during which it acts. For a constant force this simplifies to:

J = F · Δt

where F is the net force in newtons and Δt is the time interval in seconds. The result has units of N·s (newton-seconds), which are identical to kg·m/s.

Newton’s second law in its original form says that force equals the rate of change of momentum: F = Δp / Δt. Rearranged, this gives the impulse-momentum theorem:

J = Δp = m · v_f − m · v_i

Impulse equals the change in momentum. This is why a 70 kg person decelerating from 10 m/s to rest always receives the same 700 N·s impulse — the only question is how large the force is and over what time. Airbags and crumple zones exploit this by maximising Δt, which minimises the peak force.

Solve-for-variable modes

The calculator supports seven modes, hiding inputs that are not needed:

Impulse from force — J = F · Δt. Give the applied force and contact time.
Impulse from momentum change — J = m·v_f − m·v_i. Give mass and both velocities.
Force — F = J / Δt. Useful for estimating peak collision forces.
Time interval Δt — Δt = J / F. How long must a pad stay in contact?
Final velocity v_f — v_f = (J + m·v_i) / m. What speed does a kicked ball reach?
Initial velocity v_i — v_i = (m·v_f − J) / m. Back-calculate the incoming speed.
Mass — m = J / (v_f − v_i). Infer mass from measured velocity change and impulse.

Worked example

A 0.145 kg baseball leaves a bat at 40 m/s (in the opposite direction to its arrival at −30 m/s). The bat is in contact for 1 ms (0.001 s).

Using the momentum-change method:

J = m · v_f − m · v_i
J = 0.145 × 40 − 0.145 × (−30)
J = 5.8 − (−4.35)
J = 10.15 N·s

The average force exerted by the bat:

F = J / Δt = 10.15 / 0.001 = 10,150 N  (about 1 tonne-force)

That enormous force lasts only a millisecond — a vivid demonstration of why short contact times produce extreme forces for a given impulse.

Formula reference

Rearrangement	Formula	Solve when you know
Impulse (force path)	J = F · Δt	Force and contact time
Impulse (momentum path)	J = m·v_f − m·v_i	Mass and both speeds
Force	F = J / Δt	Impulse and contact time
Contact time	Δt = J / F	Impulse and force
Final velocity	v_f = (J + m·v_i) / m	Impulse, mass, initial speed
Initial velocity	v_i = (m·v_f − J) / m	Impulse, mass, final speed
Mass	m = J / (v_f − v_i)	Impulse and both speeds

All calculations are in SI units. The momentum snapshot panel (visible in momentum modes) shows p_i, p_f, and Δp alongside the main result so you can cross-check both sides of the theorem at once.