Notes : Collisions in One Dimension (1D Collision) - Class 11 Physics
Collisions in One Dimension (1D Collision)
Introduction
A one-dimensional collision is a collision in which the motion of both bodies before and after collision takes place along the same straight line.
- Completely Inelastic Collision
- Elastic Collision
1. Completely Inelastic Collision
In a completely inelastic collision, the colliding bodies stick together and move with a common velocity after collision.
Given
Mass m₁ moves with initial velocity u and mass m₂ is initially at rest.
\[ \theta_1=\theta_2=0 \]
Conservation of Momentum
\[ m_1 u=(m_1+m_2)v \]
Therefore,
\[ v=\frac{m_1}{m_1+m_2} u \]
Final Velocity:
\[ \boxed{v=\frac{m_1}{m_1+m_2} u} \]
Loss of Kinetic Energy
Initial kinetic energy:
\[ K_i=\frac{1}{2}m_1 u^{2} \]
Final kinetic energy:
\[ K_f=\frac{1}{2}(m_1+m_2)v^{2} \]
Loss in kinetic energy:
\[ \Delta K = \frac{1}{2}m_1 u^{2} -\frac{1}{2}(m_1+m_2)v^{2} \]
Substituting
\[ v=\frac{m_1}{m_1+m_2} u \]
we obtain
\[ \Delta K = \frac{1}{2} \frac{m_1m_2}{m_1+m_2} u^{2} \]
Loss of Kinetic Energy:
\[ \boxed{ \Delta K= \frac{1}{2} \frac{m_1m_2}{m_1+m_2} u^{2} } \]
Since $\Delta K > 0$, kinetic energy is lost during collision.
2. Elastic Collision
In an elastic collision:
- Momentum is conserved.
- Kinetic energy is conserved.
Momentum Conservation
\[ m_1 u = m_1 v_{1} + m_2 v_{2} \]
Kinetic Energy Conservation
\[ m_1 u^{2} = m_1 v_{1}^{2} + m_2 v_{2}^{2} \]
Derivation
From momentum and kinetic energy conservation equations,
\[ v_{2}(u-v_{1}) = (u-v_{1})(u+v_{1}) \]
Hence,
\[ v_{2}= u+v_{1} \]
Substituting into the momentum equation,
\[ v_{1} = \frac{m_1-m_2}{m_1+m_2} u \]
and
\[ v_{2} = \frac{2m_1}{m_1+m_2} u \]
Final Velocity Formulae
Velocity of first mass:
\[ \boxed{ v_{1} = \frac{m_1-m_2}{m_1+m_2} u } \]
Velocity of second mass:
\[ \boxed{ v_{2} = \frac{2m_1}{m_1+m_2} u } \]
Special Cases
Case I: Equal Masses
If
\[ m_1=m_2 \]
then
\[ v_{1}=0 \]
\[ v_{2}=u \]
The first body comes to rest and transfers its entire velocity to the second body.
Case II: One Mass Dominates
If
\[ m_2 \gg m_1 \]
then
\[ v_{1}\approx -v_{1} \]
\[ v_{2}\approx 0 \]
The heavier body remains almost unaffected while the lighter body rebounds with nearly the same speed in the opposite direction.
Key Points
- Momentum is conserved in every collision.
- Kinetic energy is conserved only in elastic collisions.
- In a completely inelastic collision, bodies stick together.
- Maximum kinetic energy loss occurs in a completely inelastic collision.
- For equal masses in an elastic collision, velocities are exchanged.
FAQ
Q1. What is a one-dimensional collision?
A collision in which all velocities before and after collision lie along the same straight line.
Q2. Is momentum conserved in an inelastic collision?
Yes. Momentum is always conserved.
Q3. Is kinetic energy conserved in a completely inelastic collision?
No. A part of kinetic energy is lost.
Q4. What happens in a completely inelastic collision?
The colliding bodies stick together and move with a common velocity.
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