PHYSICS / COLLISIONS
Calculate both final velocities in a 1D elastic collision from two masses and two signed initial velocities, with worked substitutions and conservation checks. Review the symbolic equation, substituted SI values, assumptions, and result together.
SIGNED-AXIS COLLISION LEDGER
This page is built for the part of elastic-collision problems people usually miss: the sign on each velocity. In the reviewed default example, a 2 kg cart at +5 m/s meets a 3 kg cart at -1 m/s, rebounds at -2.2 m/s, and sends the second cart away at +3.8 m/s.
Every speed on this calculator must live on one shared axis. Positive means motion in your chosen forward direction and negative means motion back along that same line. The negative final speed in the example is therefore a real rebound, not a formatting choice.
Worked default example, line by line
Enter both masses as positive values and both initial velocities as signed values on one agreed axis. The main result card returns object one's final velocity, and the first metric underneath it returns object two's final velocity. A negative final value means that object leaves opposite to your chosen positive direction.
Nirmion evaluates the closed-form one-dimensional elastic-collision equations from the linked OpenStax source: v1=[(m1-m2)u1+2m2u2]/(m1+m2) and v2=[2m1u1+(m2-m1)u2]/(m1+m2). The worked ledger verifies the returned pair three ways: total momentum stays at 7 kg m/s, total kinetic energy stays at 26.5 J, and the separation speed matches the 6 m/s approach speed.
Use this only for two bodies colliding on one line with negligible external impulse and no loss of kinetic energy to deformation, sound, heat, friction, or rotation. It is not a crash-reconstruction tool, not a two-dimensional impact solver, and not a substitute for measured test data. If the bodies move together after impact, switch to the related Perfectly Inelastic Collision Calculator.
COMMON QUESTIONS
It means the object leaves in the direction opposite to your chosen positive axis. In the default example, the first cart starts at +5 m/s and rebounds at -2.2 m/s, so the sign change tells you it turned around.
For an ideal one-dimensional elastic collision, the closing speed before impact equals the separation speed after impact. Here 5 - (-1) and 3.8 - (-2.2) both equal 6 m/s, which is a quick check that the example remains inside the elastic model.
Use that related tool when the two bodies stick together after impact or when you only need the shared final velocity of a sticking collision. This page assumes the bodies separate and keep total kinetic energy unchanged.
Use boundary
Calculate both final velocities in a 1D elastic collision from two masses and two signed initial velocities, with worked substitutions and conservation checks. The page evaluates the displayed equation from your supplied values and presents both symbolic and substituted KaTeX working so the arithmetic can be checked.
v1=[(m1-m2)u1+2m2u2]/(m1+m2)
What you provide
What you receive
Use boundary
Momentum and kinetic energy are conserved, bodies do not rotate, and external impulse is negligible.
Use internally consistent units and retain extra precision when carrying the result into another calculation.
Reviewed reference factors