# One-dimensional collisions

One-dimensional collision between body A and body B

**collision**or

**collision**when the interaction takes place in a relatively short time interval during which the effect of external forces can be neglected and, both before and after this time interval, the force of interaction between the bodies is zero or negligible.

Let’s look at the figure above: it shows us a case of one-dimensional collision between two bodies A and B, of mass *m _{A} in _{B}* , which separate after the collision. Let’s suppose that the values of

*m*are known and that we want to determine the values of

_{A}, m_{B}, v_{A}and v_{B}*v’*. The first step is to consider the conservation of the momentum of the system:

_{A}and v’_{B}Because the equation has two unknowns, the data are not enough to solve the one-dimensional collision problem. Isaac Newton, however, discovered, through his experiments, a relationship between the speeds of bodies before and after the collision. His relationship was as follows:

Newton called the letter e the *coefficient of restitution* . Therefore, in the above equation, the difference *v’ _{A} – v’ _{B}* is the velocity of A with respect to B after the collision, and the difference

*v*is the velocity of A with respect to B before the collision. However, these differences will have opposite signs, because before the collision the bodies approach and after the collision the bodies move away. In this way, we have:

_{A}– v_{B}And so the minus sign in the above equation is placed for e terms > 0.