KAHIBARO
Table of Contents
Motion Along a Single Line
A one dimensional collision is a collision in which all motion happens along the same straight line. This is the simplest kind of collision to analyze because each object needs only one signed velocity value. A velocity to the right may be taken as positive, and a velocity to the left as negative.
In this chapter, the key idea is that after choosing a direction as positive, every velocity and momentum must keep its sign. This lets us write collision equations correctly and avoid common mistakes.
A one dimensional collision can involve objects moving toward each other, one object catching another from behind, or one object initially at rest. In every case, the motion before and after the collision lies on the same axis.
Describing the Collision
Suppose two objects of masses $m_1$ and $m_2$ move along the $x$ axis. Before the collision their velocities are $u_1$ and $u_2$, and after the collision their velocities are $v_1$ and $v_2$.
The total momentum before collision is
$$
p_{\text{before}} = m_1 u_1 + m_2 u_2
$$
and the total momentum after collision is
$$
p_{\text{after}} = m_1 v_1 + m_2 v_2
$$
For an isolated system, momentum is conserved, so
$$
m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2
$$
This single equation always applies to one dimensional collisions, no matter whether the collision is elastic or inelastic.
For any one dimensional collision in an isolated system,
$$
m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2
$$
Use signed velocities. A negative velocity means motion in the opposite direction.
Why Signs Matter
Because motion occurs on one line, direction is carried by the sign of velocity. This makes one dimensional collision problems much easier than two dimensional ones, but only if signs are handled carefully.
If right is chosen as positive, then a leftward velocity must be written as negative. For example, if two objects move toward each other, one might have $u_1 = +4\ \text{m/s}$ and the other $u_2 = -2\ \text{m/s}$.
A negative result after solving means the object finally moves in the negative direction. That is a physical answer, not a mistake.
Types of One-Dimensional Collision Outcomes
The same straight line setup can produce different outcomes depending on the nature of the collision.
In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, the objects stick together and move with a common final velocity.
Those categories are developed in their own chapters, but in one dimension the main difference is that all these cases are described using the same line axis and signed quantities.
Relative Speed in One Dimension
One dimensional collisions are especially convenient because the speeds of approach and separation can be compared directly.
Before collision, the relative speed of approach is the difference in velocities along the line. After collision, the relative speed of separation is also found from the velocity difference.
For elastic collisions, a very useful relation is
$$
u_1 - u_2 = -(v_1 - v_2)
$$
which is often written as
$$
u_1 - u_2 = v_2 - v_1
$$
This says that in a one dimensional elastic collision, the relative speed of approach equals the relative speed of separation.
For a one dimensional elastic collision,
$$
u_1 - u_2 = v_2 - v_1
$$
This relation is valid only for elastic collisions.
Common Special Cases
Some one dimensional collision problems appear so often that it helps to recognize their structure immediately.
Second Object Initially at Rest
If the second object starts at rest, then $u_2 = 0$. The momentum equation becomes
$$
m_1 u_1 = m_1 v_1 + m_2 v_2
$$
This case is common in laboratory examples, such as a moving cart striking a stationary cart.
Perfectly Inelastic Collision
If the objects stick together after impact, then their final velocities are equal:
$$
v_1 = v_2 = v
$$
Then momentum conservation gives
$$
m_1 u_1 + m_2 u_2 = (m_1 + m_2)v
$$
so the shared final velocity is
$$
v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}
$$
Equal Masses in an Elastic Collision
If $m_1 = m_2$ and the collision is elastic, the two objects exchange velocities:
$$
v_1 = u_2, \qquad v_2 = u_1
$$
This result is a classic feature of one dimensional elastic collisions.
In a one dimensional elastic collision of equal masses,
$$
v_1 = u_2, \qquad v_2 = u_1
$$
The objects exchange velocities.
Visualizing the Motion
A collision on one line can be sketched simply as motion along the $x$ axis.
In this sketch, the first object moves to the right and the second moves to the left. If right is positive, then $u_1 > 0$ and $u_2 < 0$.
Solving One-Dimensional Collision Problems
The usual method is to define a positive direction, assign signs to all initial and final velocities, write the momentum equation, and then add any extra condition that belongs to the type of collision.
For example, in an elastic collision you use momentum conservation together with the relative speed relation. In a perfectly inelastic collision you use momentum conservation together with the condition that both final velocities are equal.
Because all motion is on one line, the algebra is often simpler than in higher dimensions.
Summary Table
| Situation | Key equation |
|---|---|
| Any isolated one dimensional collision | $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$ |
| Elastic collision | $u_1 - u_2 = v_2 - v_1$ |
| Perfectly inelastic collision | $v_1 = v_2$ |
| Perfectly inelastic final speed | $v = \dfrac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$ |
| Equal masses, elastic collision | $v_1 = u_2,\ v_2 = u_1$ |
Final Remarks
One dimensional collisions are important because they show the core structure of collision physics in its simplest form. Everything happens along one axis, so direction is handled by signs, and momentum conservation becomes a single algebraic equation. Once this is understood, more detailed collision types become much easier to study.
The most common mistake in one dimensional collisions is ignoring direction.
Always choose a positive direction first, then write every velocity with the correct sign.