Notes : Collisions in Two Dimensions - Class 11 Physics

Collisions in Two Dimensions

A two-dimensional collision occurs when two bodies collide and move in a plane after collision. Linear momentum is conserved in such collisions. Since momentum is a vector quantity, its conservation must be applied separately along the x-axis and y-axis.

Consider

• Mass m₁ moving initially with velocity v₁ᵢ.
• Mass m₂ initially at rest.
• After collision:
  • m₁ moves with velocity v₁f at angle θ₁.
  • m₂ moves with velocity v₂f at angle θ₂.

Conservation of Linear Momentum

Along the x-axis

$$ m_1v_{1i}=m_1v_{1f}\cos\theta_1+m_2v_{2f}\cos\theta_2 $$

Along the y-axis

$$ 0=m_1v_{1f}\sin\theta_1-m_2v_{2f}\sin\theta_2 $$

Unknown Quantities

Usually, the known quantities are:

$$ \{m_1,\;m_2,\;v_{1i}\} $$

The unknown quantities are:

$$ \{v_{1f},\;v_{2f},\;\theta_1,\;\theta_2\} $$

Thus, there are four unknowns but only two momentum equations.

Special Case: One-Dimensional Collision

If

$$ \theta_1=\theta_2=0 $$

the collision reduces to a one-dimensional collision.

Elastic Collision

If the collision is elastic, kinetic energy is also conserved:

$$ \frac{1}{2}m_1v_{1i}^{2} = \frac{1}{2}m_1v_{1f}^{2} + \frac{1}{2}m_2v_{2f}^{2} $$

This provides one additional equation.

Solvability of the Problem

Even after applying conservation of kinetic energy:

• Number of unknowns = 4
• Number of equations = 3

Therefore, one additional quantity (usually an angle such as θ₁) must be known experimentally.

If

$$ \{m_1,\;m_2,\;v_{1i},\;\theta_1\} $$

are known, then

$$ \{v_{1f},\;v_{2f},\;\theta_2\} $$

can be determined using the above equations.

Key Points

• Momentum is conserved separately along the x-axis and y-axis.
• The collision takes place entirely in the x-y plane.
• For elastic collisions, kinetic energy is also conserved.
• Two-dimensional collision problems generally require one additional measured quantity.
• One-dimensional collision is a special case when both scattering angles are zero.

FAQ

Q1. Why are two momentum equations required?

Because momentum is a vector quantity and must be conserved separately along the x and y directions.

Q2. Is kinetic energy always conserved?

No. Kinetic energy is conserved only in elastic collisions.

Q3. Why can't the problem be solved using only momentum equations?

There are four unknowns but only two momentum equations.

Q4. What additional information is needed?

Usually one of the scattering angles, such as θ₁, is measured experimentally.

Q5. What happens when θ₁ = θ₂ = 0?

The collision becomes one-dimensional.

Quiz

1. Which quantity is always conserved in a collision?
   (a) Kinetic Energy
   (b) Momentum
   (c) Velocity
   (d) Force
   Answer: (b) Momentum
2. In a two-dimensional collision, momentum is conserved along:
   (a) x-axis only
   (b) y-axis only
   (c) Both x-axis and y-axis
   (d) Neither axis
   Answer: (c) Both x-axis and y-axis
3. Kinetic energy is conserved in:
   (a) Inelastic collision
   (b) Elastic collision
   (c) Perfectly inelastic collision
   (d) None
   Answer: (b) Elastic collision
4. How many unknowns are present in the general two-dimensional collision discussed?
   (a) 2
   (b) 3
   (c) 4
   (d) 5
   Answer: (c) 4
5. A two-dimensional collision becomes one-dimensional when:
   (a) m₁ = m₂
   (b) v₂f = 0
   (c) θ₁ = θ₂ = 0
   (d) v₁f = 0
   Answer: (c) θ₁ = θ₂ = 0