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The linear impulse-momentum principle states that the time integral of net force equals the change in linear momentum: ∫F dt = mv₂ − mv₁. This method is most efficient when force is given as a function of time or when an average force must be found from a velocity change over a known time interval. If the net external impulse is zero in a given direction, linear momentum is conserved in that direction. The principle applies independently in each coordinate direction.
Use impulse-momentum when time is explicitly involved in the problem. Choose between work-energy and impulse-momentum based on whether the problem involves displacement (work-energy) or time interval (impulse-momentum). Apply conservation of momentum only when net external impulse is truly zero.
You already know Newton's second law: F = ma. Written out, this is F = m(dv/dt). Multiply both sides by dt and integrate: ∫F dt = m·v₂ − m·v₁. That integral of force over time is the impulse, and the right-hand side is the change in linear momentum (L = mv). This impulse-momentum principle is not a new law — it is Newton's second law rewritten for problems where time is the natural variable rather than displacement.
The key decision in dynamics is choosing your method before you start. Work-energy relates force to displacement: ΔKE = ∫F·ds. Impulse-momentum relates force to time: ΔL = ∫F dt. If the problem gives you a force as a function of time (like a rocket thrust profile) or asks for an average force over a known time interval, reach for impulse-momentum. If it gives displacement or asks about speed at a certain position, reach for work-energy. Mixing the methods on the same problem is a common error. Pick the one that matches what is known and what is unknown.
When the force varies with time, the impulse is the area under the F-vs-t curve. For constant force it simplifies to F·Δt. When the problem states that an average force acted over a known time (as in a bat striking a ball), you can find F_avg = ΔL/Δt directly without knowing any trajectory details. This is exactly how engineers measure impact forces: measure the velocity change (with high-speed cameras or accelerometers) and the contact duration (with force plates or strain gauges), and the average impact force follows immediately.
Because momentum is a vector, the principle applies independently in each coordinate direction. Write separate equations for x and y (and z in 3D): (∫F_x dt = m·v₂ₓ − m·v₁ₓ), and so on. This is especially useful in oblique impacts or problems where a component of external impulse is zero in one direction but not another. If there is no external x-impulse, x-momentum is conserved — you get a free equation. If gravity acts vertically during the time interval, it contributes an impulse of mg·Δt in the y-direction; this is often small for very brief impacts (and can be neglected) but is significant for slower interactions like a ball rolling or a rocket firing over seconds.