Collisions in One Dimension

Collisions in One Dimension

When two objects collide, the forces they exert on each other are internal to the two-body system. If no external force acts during the collision, the total momentum of the system is conserved. This is the bedrock principle. But what about kinetic energy? That depends entirely on the nature of the collision.

Elastic and Inelastic Collisions

A collision where the total kinetic energy of the system is conserved is called an elastic collision. A collision where some kinetic energy is transformed into other forms (heat, sound, deformation) is called an inelastic collision. In a perfectly inelastic collision, the two bodies stick together after impact and move with a common velocity.

Elastic Collision in One Dimension

Consider two bodies of masses $m_1$ and $m_2$ moving along a straight line with initial velocities $u_1$ and $u_2$ respectively. Let $v_1$ and $v_2$ be their velocities after the collision. We assume the collision is elastic and one-dimensional.

From conservation of momentum:

From conservation of kinetic energy:

We have two equations and two unknowns ($v_1$ and $v_2$). To solve them efficiently, we rearrange.

From (1):

From (2):

Divide equation (4) by equation (3) — provided $u_1 \neq v_1$ and $v_2 \neq u_2$, which is true for a genuine collision:

Now we can solve for $v_1$ and $v_2$. From (5), $v_2 = u_1 + v_1 - u_2$. Substitute into (1):

Similarly, substituting $v_1 = v_2 + u_2 - u_1$ from (5) into (1) gives:

These are the general formulas for velocities after a one-dimensional elastic collision.

Special Cases of Elastic Collision

Case 1: Equal masses ($m_1 = m_2$)

Substitute $m_1 = m_2 = m$ into the formulas:

Case 2: A very heavy body colliding with a very light body at rest

Let $m_1 \gg m_2$ and $u_2 = 0$. The formulas simplify. For $v_1$:

The heavy body continues with almost unchanged velocity.

For $v_2$:

Case 3: A very light body colliding with a very heavy body at rest

Let $m_1 \ll m_2$ and $u_2 = 0$. Then:

Inelastic Collision in One Dimension

In an inelastic collision, kinetic energy is not conserved. Momentum is still conserved. For a perfectly inelastic collision, the two bodies stick together and move with a common velocity $v$ after collision.

From momentum conservation:

The loss in kinetic energy is:

Substituting $v$ and simplifying gives:

$\Delta K = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (u_1 - u_2)^2$ …