Coefficient of Restitution Collision Calculator

Understanding the Coefficient of Restitution

The coefficient of restitution, commonly denoted by the symbol $e$ , quantifies how “bouncy” a collision is. When two objects collide along a single line, they compress momentarily and then separate. The ratio of their relative speed after the collision to the relative speed before the collision defines $e$ . Specifically, $e = \frac{v_{2} - v_{1}}{u_{1} - u_{2}}$ , where $u_{1}$ and $u_{2}$ are the initial velocities of masses $m_{1}$ and $m_{2}$ respectively, while $v_{1}$ and $v_{2}$ are their velocities after impact. Values of $e$ range from 0 to 1 for physical materials, with $e = 1$ corresponding to a perfectly elastic collision that conserves kinetic energy, and $e = 0$ representing a perfectly inelastic collision in which the bodies stick together. Intermediate values indicate partial energy loss to deformation, heat, or sound.

Deriving the Final Velocity Formulas

To determine the velocities after a collision, one combines momentum conservation with the definition of $e$ . Conservation of linear momentum for a closed system requires $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ . Substituting the restitution relationship for one of the velocities and solving simultaneously yields explicit expressions for both $v_{1}$ and $v_{2}$ . After algebraic manipulation, the results are $v_{1} = \frac{m_{1} u_{1} + m_{2} u_{2} - m_{2} e (u_{1} - u_{2})}{m_{1} + m_{2}}$ and $v_{2} = \frac{m_{1} u_{1} + m_{2} u_{2} + m_{1} e (u_{1} - u_{2})}{m_{1} + m_{2}}$ . The calculator implements these formulas, allowing students to explore how varying the masses, incoming speeds, and restitution coefficient influence the rebound.

Physical Meaning and Material Properties

Different materials exhibit characteristic restitution coefficients. Billiard balls on a felt table may have $e$ around 0.9, while clay clumps possess values near 0. Understanding $e$ helps engineers select materials for desired energy dissipation. For instance, car bumpers are designed to lower $e$ to absorb impact energy, protecting passengers. Sporting equipment often seeks higher $e$ to maximize rebound, as in superballs or golf club faces. The table below lists typical coefficients for common materials colliding with rigid surfaces.

Material	Approximate e
Steel on Steel	0.6–0.9
Rubber on Concrete	0.8
Wood on Wood	0.4–0.5
Glass on Glass	0.94
Clay on Steel	0.0–0.2

Range of Applicability

The one‑dimensional treatment assumes objects move along a single straight line and interact instantaneously. Real collisions may involve rotation, deformation, or off‑center impacts that require more complex modeling. Nevertheless, the simplified model captures essential features of many practical problems, especially when objects collide head‑on or when one mass greatly exceeds the other. The coefficient of restitution remains a useful scalar measure even in more complex situations, though vector analysis becomes necessary for accurate predictions.

Worked Example

Consider a ball of mass $0.5$ kg moving at $4$ m/s that collides with a stationary ball of mass $1$ kg. Suppose the coefficient of restitution is $0.8$ . Substituting into the formulas above gives $v_{1} = \frac{0.5×4 + 1×0 - 1×0.8×(4 - 0)}{1.5} = - 0.533 m/s$ and $v_{2} = \frac{0.5×4 + 1×0 + 0.5×0.8×(4 - 0)}{1.5} = 2.13 m/s$ . The lighter ball reverses direction with a small speed, while the heavier ball moves forward at a moderate pace. Changing $e$ to 1 would yield more energetic rebounds, while $e = 0$ would make both balls move together at $\frac{0.5×4}{1.5} = 1.33$ m/s.

Energy Considerations

Although momentum is always conserved in isolated systems, kinetic energy is not necessarily preserved. The fractional kinetic energy after the collision compared to before depends on $e$ . For equal masses and one object initially at rest, the ratio simplifies to $e^{2}$ . Thus, even an apparently small reduction in $e$ can significantly decrease kinetic energy. Engineers design safety devices by choosing materials and geometries that yield low $e$ , converting kinetic energy into deformation work that protects occupants.

Laboratory Measurement

Determining the coefficient of restitution experimentally is straightforward. One can drop a ball from a known height onto a hard surface and measure the rebound height. The ratio of rebound speed to impact speed equals the square root of the ratio of heights, since gravitational potential energy converts fully to kinetic energy during free fall. Therefore, $e = \sqrt{\frac{h_{r}}{h_{d}}}$ , where $h_{d}$ is the drop height and $h_{r}$ is the rebound height. Precision requires minimizing air resistance and ensuring a rigid surface. For colliding carts on a track, photogates or motion sensors capture velocities before and after impact, offering high accuracy.

Dependence on Impact Conditions

Real materials exhibit restitution coefficients that vary with impact speed, temperature, and surface properties. At very low speeds, viscoelastic effects may cause $e$ to drop as internal friction dissipates energy. At high speeds, plastic deformation or cracking can occur, dramatically reducing $e$ . Surface roughness and lubrication influence the duration of contact and energy losses. Therefore, quoted $e$ values typically specify the conditions under which they were measured. In some sports, rule books define testing protocols to ensure consistent behavior of equipment.

Applications in Sports and Engineering

The coefficient of restitution plays a central role in sports. Baseball bat manufacturers tune $e$ to maximize the “trampoline effect,” enabling balls to leave the bat at higher speeds. In golf, regulations limit club face flexibility to control driving distance. In automotive crash testing, engineers analyze collision data to characterize $e$ for components and entire vehicles. Robotics researchers utilize restitution to simulate contact dynamics and design grippers that avoid unwanted bouncing. In planetary science, the outcome of asteroid collisions, whether they fragment or accrete, depends partly on restitution properties of regolith and rock.

Worked Table of Scenarios

The following table illustrates how varying the coefficient of restitution affects collision outcomes for equal masses, with one mass initially stationary and the other moving at $5$ m/s.

Coefficient e	v₁ after (m/s)	v₂ after (m/s)	Final Kinetic Energy / Initial
1.0	0	5	1.00
0.8	-1	4	0.64
0.5	-2	3	0.25
0.0	-2.5	2.5	0.00

Limitations of the Model

The idealized formulas neglect rotational kinetic energy, friction, and multi-dimensional motion. If bodies possess spin or collide off-center, angular momentum considerations become necessary, leading to changes in rotation as well as translation. Soft bodies may experience extended contact times with complex force histories that defy simple restitution modeling. Nevertheless, the coefficient remains a convenient scalar parameter for many practical problems where detailed force measurements are unavailable.

Historical Context

Sir Isaac Newton introduced the concept of restitution in his landmark 1687 work, Philosophiæ Naturalis Principia Mathematica. He recognized that real collisions do not conserve kinetic energy and devised the restitution coefficient to account for the observed rebound behavior. Subsequent scientists refined measurements and linked restitution to material properties. Today, the coefficient is a staple in physics education and engineering practice, bridging theoretical mechanics and real-world imperfections.

Using the Calculator

To employ this calculator, enter the masses and initial velocities of two bodies along with an estimated restitution coefficient. Positive velocities indicate motion to the right, while negative values denote leftward motion. After clicking the compute button, the tool applies the formulas derived above to output final velocities. The numerical engine handles any combination of masses and speeds, including cases where one object is initially at rest or where both approach each other. Experimenting with different values builds intuition for collision dynamics and energy dissipation.

Conclusion

The coefficient of restitution succinctly characterizes how dramatically objects bounce apart after colliding. By combining this empirical measure with conservation laws, one can predict post-collision velocities in many situations. The extensive explanation provided here explores the physics underlying the concept, offering derivations, examples, and real-world applications. Mastery of restitution equips students, engineers, and enthusiasts to analyze impacts ranging from toy balls to planetary encounters.