Momentum
Impulse as an integral, \(J=\int F\,dt\), for the same reason work needed one in the previous unit: the moment force varies with time, "force times time" stops being valid, and computing the actual change in momentum means integrating. The impulse-momentum theorem itself falls directly out of \(F=dp/dt\) — force is defined as the rate of change of momentum, and integrating that definition over time is the entire derivation.
What you'll learn
- Compute impulse from a time-dependent force by integrating, rather than assuming force is constant
- Derive the impulse-momentum theorem directly from F=dp/dt
- Find a time-varying force from a given momentum function by differentiating
- Apply conservation of momentum to collisions, unchanged from the algebra-based treatment
1. Impulse as an Integral
Impulse is usually introduced as \(J=F\Delta t\), valid only when force is constant. Real forces during a collision or a push are rarely constant, so the general definition is \(J=\int F(t)\,dt\) — exactly parallel to how work needed an integral once force stopped being constant in position.
2. The Impulse-Momentum Theorem
Newton's second law in its most fundamental form is \(F=dp/dt\) — force is defined as the rate momentum changes, and \(F=ma\) is just the special case where mass stays fixed (so \(dp/dt=m\,dv/dt=ma\)). Integrating \(F=dp/dt\) over time gives \(\int F\,dt = \Delta p\) directly: the impulse-momentum theorem isn't a separate law, it's this definition integrated.
3. Conservation of Momentum in Collisions
None of the calculus above changes how conservation of momentum works in a collision: as long as the net external force on the colliding system is zero (or negligible over the short collision time), total momentum before equals total momentum after. This holds regardless of how complicated the internal forces between the colliding objects actually are during impact — which is exactly the point of using momentum conservation instead of trying to track those forces directly.
Key equations
- J = ∫F(t)dt — The general definition of impulse — reduces to FΔt only when F is constant.
- F = dp/dt — Newton's second law in its most general form — force is the rate of change of momentum, valid even if mass isn't constant (though this course keeps mass fixed).
- J = Δp = ∫F dt — The impulse-momentum theorem — integrating F=dp/dt over time directly gives this, the same way integrating F=ma over distance gave the work-energy theorem.