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Mechanical

Direct impact with restitution

Velocity of body A after a collision along the line of impact.

Ref: MEC-17
Rev: v1
Field: Mechanical
Result: v_A2 [m/s]

e = \frac{v_{B2} - v_{A2}}{v_{A1} - v_{B1}}

Signed velocities along the line of impact, with A approaching B from behind (v_A1 > v_B1). Combining the restitution relation with conservation of momentum gives v_A2 = [m_Av_A1 + m_Bv_B1 − m_Be(v_A1 − v_B1)] / (m_A + m_B).

After a collision the bodies separate more slowly than they approached; the restitution coefficient says by how much.

Calculator

Solve for

v_A2Velocity of A afterComputed

m/s

m_AMass of A

0.1100

m_BMass of B

0.1100

v_A1Velocity of A before

m/s

150

v_B1Velocity of B before

m/s

020

eCoefficient of restitution

Velocity of B after4.56 m/s
Kinetic energy lost6.144 J

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Velocity of A after (m/s) current value 2.16 m/s

v_B1Velocity of B beforeKey driver

0.2419.4

m_AMass of A

−0.1945.81

m_BMass of B

−0.2755.7

eCoefficient of restitution

1.23.6

v_A1Velocity of A before

1.963.92

Momentum is conserved in any collision; kinetic energy is lost unless e = 1.
With e = 0 the bodies move together after impact.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m/s)	Local step	Change in result
v_B1 Velocity of B before	0 to 20 m/s	0.24	19.44	19.2	+10%	+8.89 %
m_A Mass of A	0.1 to 100 kg	−0.1935	5.814	6.007	+10%	+6.84 %
m_B Mass of B	0.1 to 100 kg	−0.2745	5.695	5.97	+10%	−6.71 %
e Coefficient of restitution	0 to 1	1.2	3.6	2.4	+undefined%	—
v_A1 Velocity of A before	1 to 50 m/s	1.96	3.92	1.96	+10%	+1.11 %

Sweep of Coefficient of restitution

Coefficient of restitution	Velocity of A after (m/s)
0	3.6
0.125	3.3
0.25	3
0.375	2.7
0.5	2.4
0.625	2.1
0.75	1.8
0.875	1.5
1	1.2

Sweep one variable

Vary

Velocity of A after (m/s)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

v_A2

Velocity of A after

Result · m/s

What the equation gives you. Everything else feeds it.

m_A

Mass of A

kg · typical 0.1 to 100

Mass of body A. A heavier A keeps more of its speed.

m_B

Mass of B

kg · typical 0.1 to 100

Mass of body B. A heavier B slows A more.

v_A1

Velocity of A before

m/s · typical 1 to 50

Velocity of A before impact. Raises v_A2.

v_B1

Velocity of B before

m/s · typical 0 to 20

Velocity of B before impact. Raises v_A2.

Coefficient of restitution

· typical 0 to 1

0 is perfectly plastic, 1 perfectly elastic. A larger e makes A rebound more.

Worked example with the values above

Total momentum: 2×6 + 3×2 = 18 kg·m/s.
Restitution term: m_Be(v_A1 − v_B1) = 7.2.
v_A2 = (18 − 7.2) / 5 = 2.16 m/s.

Assumptions and limits

Collision along the line of impact with negligible external impulse.
Constant e.

Common mistakes

Forgetting velocity signs.
Applying restitution to the wrong pair of velocities.

Statistics and uncertainty

The restitution coefficient is usually the least certain input and depends on speed and materials.

Where it is used

Impact and collision analysis.
Arrester and bumper design.

Maths you need

Momentum: p = mv
Relative velocity: closing and separating speeds

Related formulas

Practise this formula