Coefficient of Restitution Calculator
Calculate the coefficient of restitution from the relative speeds of two objects before and after a collision, where 1 is elastic and 0 inelastic.
Coefficient of Restitution Calculator
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Formula: e = relative speed after / relative speed before
Worked example — e=0.6
Formula
e = relative speed after / relative speed before
Ratio of relative speed after to before collision. e=1 perfectly elastic, e=0 perfectly inelastic.
Worked Examples
Example 1: Ball bounce
Problem:Ball: v1=10, v2=0, v1 after=2, v2 after=8
Solution:e=(8-2)/(10-0)=0.6
Result:e=0.6
Frequently Asked Questions
What affects the coefficient of restitution?
Material (rubber≈0.8, steel≈0.6, clay≈0), temperature, and impact speed. It is not truly constant but approximately so for moderate speeds.
How is the coefficient of restitution calculated?
e = (relative speed after collision) / (relative speed before collision) = |v2' - v1'| / |v1 - v2|. A value of 1 means no kinetic energy loss; a value of 0 means the objects stick together after impact.
When would I use Coefficient of Restitution Calculator?
Sports equipment design (tennis balls, golf balls, basketball standards), automotive crash testing, industrial machinery collision analysis, and physics research all rely on measuring the coefficient of restitution to characterize material behavior.
How do I measure the coefficient of restitution with a simple drop test?
Drop the object from rest onto a heavy rigid surface and record the drop height and the rebound height. Because impact and rebound speeds follow from free fall, e is the square root of the rebound height divided by the drop height. A ball released from 1.00 m that returns to 0.36 m gives e equal to the square root of 0.36, that is 0.6, matching the velocity definition used here.
What is the difference between an elastic, an inelastic and a perfectly inelastic collision?
Momentum is conserved in every collision, but kinetic energy is not. An elastic collision has e equal to 1 and conserves kinetic energy exactly. A partially inelastic collision has e between 0 and 1 and loses energy to heat, sound and permanent deformation. In a perfectly inelastic collision e is 0, the bodies move off with a common velocity, and the energy loss is the maximum allowed by momentum conservation.
How do I find both velocities after a collision if I know e and the masses?
Solve two equations together: conservation of momentum, m1v1 + m2v2 = m1v1' + m2v2', and the restitution relation, v2' - v1' = e(v1 - v2). Two linear equations in two unknowns give each final velocity uniquely. Coefficient of Restitution Calculator works the other way, taking the four measured velocities and returning the e that describes the impact.
Why can the coefficient of restitution never be greater than 1?
A value above 1 would mean the bodies separate faster than they approached, creating kinetic energy from nothing and violating conservation of energy. Values above 1 are only ever reported when stored energy is released during contact, for example an explosive charge or a compressed spring latch, in which case the collision is no longer a passive impact and e loses its usual meaning.
How does e control the height of successive bounces?
Each bounce multiplies the rebound speed by e, and height scales with the square of speed, so the height after n bounces is the initial height times e raised to the power 2n. For e of 0.8 released from 1 m, the peaks fall to about 0.64 m, 0.41 m and 0.26 m. The bounces also get closer together in time, which is why a dropped ball ends in a rapid rattle.
Does the coefficient of restitution stay the same at every impact speed?
No. It is an empirical descriptor, not a material constant. As impact speed rises, more of the contact deformation becomes plastic rather than elastic, and viscoelastic materials dissipate more energy, so e typically falls. Temperature, surface condition and the stiffness of the struck body matter too, so quote e alongside the conditions under which it was measured.
How is the coefficient of restitution used in equipment testing?
Test protocols fix the variables that e depends on: a specified impact speed, a rigid instrumented plate or a defined striking body, and a controlled temperature and conditioning period. Rebound speed or rebound height is then measured and converted to e. Holding those conditions constant is what makes results from different laboratories comparable, since e is not an intrinsic property of the material alone.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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