Use of Newton's Law of Restitution

Use of Newton’s Law of Restitution

Newton’s Law of Restitution: Introduction

• Newton’s Law of Restitution is concerned with the velocity of objects just before and just after a collision. It can be applied to both elastic and inelastic collisions, and is a vital tool in mechanics analyses.

• The law states that the relative speed of two objects after an impact is equal to the product of their relative speed before the impact and a factor known as the ‘coefficient of restitution’. This is generally referred to using the symbol ‘e’.

The Coefficient of Restitution

• The coefficient of restitution ‘e’ is a value that indicates how ‘elastic’ a collision is. In a perfectly elastic collision (one in which no kinetic energy is lost), ‘e’ equals 1. In a perfectly inelastic collision (where the objects stick together after collision), ‘e’ equals 0.

• Importantly, regardless of the value of ‘e’, momentum is always conserved in collisions. This is often utilised to find unknown values in restitution problems.

Using Newton’s Law

• In practice, Newton’s Law of Restitution is expressed in the following formula: e = (vB2 - vA2) / (vA1 - vB1). Here, vA1 and vB1 are the speeds of objects A and B before collision, and vA2 and vB2 their speeds after collision.

• Usually in problems, you will be provided some of these values and asked to find the others. To do so, you will often need to combine the use of the restitution formula with the principle of conservation of momentum.

• The approach to solving these problems is usually as follows - find the valued that can be found directly from the problem statement. Then use the conservation of momentum to form a second equation. Finally, solve these equations simultaneously to find the unknown values.

• Note that Newton’s Law of Restitution can be applied in one or two dimensional problems. Accordingly, pay careful attention to direction indicators in problems - speeds in opposing directions should be denoted with opposite signs.

Special Case: Oblique Collisions

• In dealing with object collisions at an angle (oblique collisions), note that each object’s motion can be split into perpendicular components - one parallel and one perpendicular to the line of impact.

• The component of motion parallel to the line of impact is unaffected by the collision. However, the components perpendicular to the line can be analysed using Newton’s Law of Restitution as in simpler problems.