A two-dimensional collision is solved by treating momentum as a vector. Momentum must be conserved in both the
\(x\)- and \(y\)-directions if the external impulse during the collision is negligible.
Vector momentum conservation
For two objects:
\[
m_1\vec u_1+m_2\vec u_2=m_1\vec v_1+m_2\vec v_2
\]
Here \(\vec u_1,\vec u_2\) are the initial velocities and \(\vec v_1,\vec v_2\) are the final velocities.
In component form:
\[
m_1u_{1x}+m_2u_{2x}=m_1v_{1x}+m_2v_{2x}
\]
\[
m_1u_{1y}+m_2u_{2y}=m_1v_{1y}+m_2v_{2y}
\]
Line of centers for smooth round-body collisions
For smooth spheres or billiard balls, the collision impulse acts along the line of centers. This direction is called
the normal direction, written as \(\hat n\). The perpendicular direction is the tangential direction, written as \(\hat t\).
\[
\hat n=(\cos\phi,\sin\phi)
\]
\[
\hat t=(-\sin\phi,\cos\phi)
\]
Each velocity can be split into normal and tangential parts:
\[
u_n=\vec u\cdot \hat n,
\qquad
u_t=\vec u\cdot \hat t
\]
Restitution along the normal direction
The coefficient of restitution \(e\) describes how much of the normal relative speed is restored after impact:
\[
e=-\frac{v_{1n}-v_{2n}}{u_{1n}-u_{2n}}
\]
Common cases are:
- \(e=1\): perfectly elastic collision.
- \(0<e<1\): partially inelastic collision.
- \(e=0\): no rebound along the normal direction.
Smooth-collision final normal components
The normal components behave like a one-dimensional collision:
\[
v_{1n}
=
\frac{m_1u_{1n}+m_2u_{2n}-m_2e(u_{1n}-u_{2n})}{m_1+m_2}
\]
\[
v_{2n}
=
\frac{m_1u_{1n}+m_2u_{2n}+m_1e(u_{1n}-u_{2n})}{m_1+m_2}
\]
For smooth contact without friction, the tangential components do not change:
\[
v_{1t}=u_{1t},
\qquad
v_{2t}=u_{2t}
\]
The final vectors are then reconstructed:
\[
\vec v_1=v_{1n}\hat n+v_{1t}\hat t
\]
\[
\vec v_2=v_{2n}\hat n+v_{2t}\hat t
\]
Perfectly inelastic sticking collision
If the two objects stick together, they share one final velocity. Momentum conservation gives:
\[
\vec v_f=\frac{m_1\vec u_1+m_2\vec u_2}{m_1+m_2}
\]
Kinetic energy is not conserved in a sticking collision.
Kinetic energy
The kinetic energy before and after collision is:
\[
K_i=\frac12m_1u_1^2+\frac12m_2u_2^2
\]
\[
K_f=\frac12m_1v_1^2+\frac12m_2v_2^2
\]
The change in kinetic energy is:
\[
\Delta K=K_f-K_i
\]
A perfectly elastic collision has \(\Delta K=0\). An inelastic collision usually has \(\Delta K<0\), meaning kinetic
energy is transformed into heat, sound, deformation, or rotation.
Worked example: billiard-ball glancing collision
Suppose equal-mass billiard balls collide. Ball 1 moves at \(5.0\ \mathrm{m\,s^{-1}}\) along \(0^\circ\), ball 2 is
initially at rest, and the line of centers is at \(30^\circ\). Let \(e=1\).
Normal and tangential components.
\[
\hat n=(\cos30^\circ,\sin30^\circ)
\]
\[
u_{1n}=5\cos30^\circ\approx4.33\ \mathrm{m\,s^{-1}}
\]
\[
u_{1t}=5(-\sin30^\circ)\approx-2.50\ \mathrm{m\,s^{-1}}
\]
For equal masses with an elastic collision, the normal component transfers to ball 2 while ball 1 keeps its tangential
component.
\[
v_{1n}=0,
\qquad
v_{2n}=4.33\ \mathrm{m\,s^{-1}}
\]
\[
v_{1t}=-2.50\ \mathrm{m\,s^{-1}},
\qquad
v_{2t}=0
\]
Therefore ball 1 moves away along the tangent direction, while ball 2 moves along the line of centers. This is the
classic right-angle separation seen in ideal equal-mass billiard-ball collisions.
Formula summary
| Concept |
Formula |
Meaning |
| Vector momentum conservation |
\(m_1\vec u_1+m_2\vec u_2=m_1\vec v_1+m_2\vec v_2\) |
Total vector momentum is conserved. |
| x-momentum |
\(m_1u_{1x}+m_2u_{2x}=m_1v_{1x}+m_2v_{2x}\) |
Horizontal momentum balance. |
| y-momentum |
\(m_1u_{1y}+m_2u_{2y}=m_1v_{1y}+m_2v_{2y}\) |
Vertical momentum balance. |
| Normal unit vector |
\(\hat n=(\cos\phi,\sin\phi)\) |
Direction of the collision impulse. |
| Tangential unit vector |
\(\hat t=(-\sin\phi,\cos\phi)\) |
Direction perpendicular to the impulse. |
| Restitution |
\(e=-\frac{v_{1n}-v_{2n}}{u_{1n}-u_{2n}}\) |
Normal rebound ratio. |
| Object 1 normal final |
\(v_{1n}=\frac{m_1u_{1n}+m_2u_{2n}-m_2e(u_{1n}-u_{2n})}{m_1+m_2}\) |
Normal velocity after impact. |
| Object 2 normal final |
\(v_{2n}=\frac{m_1u_{1n}+m_2u_{2n}+m_1e(u_{1n}-u_{2n})}{m_1+m_2}\) |
Normal velocity after impact. |
| Sticking final velocity |
\(\vec v_f=\frac{m_1\vec u_1+m_2\vec u_2}{m_1+m_2}\) |
Final common velocity for perfectly inelastic sticking. |
| Kinetic energy change |
\(\Delta K=K_f-K_i\) |
Energy conserved, lost, or gained. |
In 2D collisions, always conserve vector momentum. Use kinetic energy only for elastic collisions, and use restitution along the collision normal.
