1D momentum conservation calculator

It applies total momentum before equals total momentum after, preserving velocity signs along one chosen axis. Use any consistent mass and velocity units. The result keeps the velocity unit you supplied and also reports the initial and reconstructed final momentum so you can verify the balance.

Set up a one-dimensional momentum problem

Begin by choosing one direction as positive and keep that convention for every velocity. Motion in the opposite direction must use a negative sign; a stationary object uses zero. Enter mass_1 and mass_2 in the same mass unit, then enter initial_velocity_1, initial_velocity_2, and the known final_velocity_1 in the same velocity unit. The calculator solves specifically for final_velocity_2. It does not assume that the objects stick together, rebound, or exchange equal speeds. Those behaviors depend on collision details, while conservation of total momentum supplies the single relationship needed here because one final velocity is already known. Both masses must be finite and nonzero. In ordinary physical applications masses should also be positive, although the computation enforces the requested nonzero rule rather than imposing a particular model beyond it. Before calculating, draw a small arrow for your positive axis and annotate every known velocity with a sign. This simple setup prevents the most common error: entering speed magnitudes while silently discarding direction. Units may be kilograms and metres per second, grams and centimetres per second, or another consistent pair. Do not mix mass or velocity units inside one calculation.

Understand the conservation equation

The calculation starts from m1 times v1i plus m2 times v2i equals m1 times v1f plus m2 times v2f. The two terms on the left are the objects' initial momenta, and the two terms on the right are their final momenta. Rearranging for the unknown gives v2f equals (m1 times v1i plus m2 times v2i minus m1 times v1f) divided by m2. The calculator evaluates that closed-form expression directly and deterministically. A negative answer is not an error: it means object 2 finishes moving opposite to the positive direction you selected. A zero answer means its final momentum is zero. Alongside final_velocity_2, the response gives initial_momentum and final_momentum. These totals should agree, apart from stable floating-point rounding, because the solved velocity is reconstructed into the final total. Momentum units are the chosen mass unit multiplied by the chosen velocity unit. The returned velocity uses exactly the velocity scale of the inputs. If a result looks surprising, inspect signs first, then confirm that both masses share a unit and all three known velocities share another unit.

Use the result within the model's limits

Momentum conservation applies to an isolated system or to a time interval in which external impulse is negligible. That makes the method useful for textbook collisions, carts on a low-friction track, idealized recoil, and preliminary checks of experimental observations. It does not by itself determine energy loss, collision duration, deformation, impact force, or whether a proposed event is physically achievable under additional constraints. For an elastic collision, kinetic-energy conservation adds another equation; for objects that lock together, both final velocities must be equal. This calculator intentionally does neither because you provide one final velocity and ask for the other. Treat measured inputs according to their precision: a very exact displayed value cannot repair uncertain masses or noisy velocity readings. When automating repeated calculations, validate the returned momentum totals and retain the original signed inputs in your records. Zero mass is rejected because division by mass_2 would be undefined and a zero mass_1 also falls outside the supported two-body model. The browser calculation is convenient for individual checks, while API requests cost $0.002 each for worksheets, lab pipelines, simulations, or teaching systems that need the same transparent equation applied consistently.