Elastic collision
Elastic Collision
An elastic collision is a type of collision where the total kinetic energy and the total momentum of the system are conserved. This means that after the collision, the total kinetic energy of the system is the same as it was before the collision. Elastic collisions are characterized by the objects 'bouncing' off each other without any loss of kinetic energy to other forms of energy, such as heat or sound.
Conservation Laws in Elastic Collisions
In an elastic collision, two conservation laws are at play:
- Conservation of Momentum: The total momentum of the system before the collision is equal to the total momentum of the system after the collision.
- Conservation of Kinetic Energy: The total kinetic energy of the system before the collision is equal to the total kinetic energy of the system after the collision.
Formulas
The conservation of momentum can be expressed as:
$$ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 $$
where:
- $m_1$ and $m_2$ are the masses of the two objects,
- $u_1$ and $u_2$ are the initial velocities of the two objects before the collision,
- $v_1$ and $v_2$ are the final velocities of the two objects after the collision.
The conservation of kinetic energy can be expressed as:
$$ \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 $$
Differences Between Elastic and Inelastic Collisions
| Aspect | Elastic Collision | Inelastic Collision |
|---|---|---|
| Kinetic Energy | Conserved | Not conserved; some is converted to other forms |
| Momentum | Conserved | Conserved |
| Deformation | Temporary; objects restore their shape | Permanent; objects may stick together or deform |
| Sound/Heat Generation | Minimal or none | Often significant |
| Equation Usage | Both conservation of momentum and kinetic energy apply | Only conservation of momentum applies |
Examples
Example 1: Collision of Two Billiard Balls
Consider two billiard balls, A and B, with equal mass $m$. Ball A is moving with velocity $u$ towards ball B, which is at rest. After the collision, ball A comes to rest and ball B moves with velocity $u$. This is an example of an elastic collision where the kinetic energy and momentum are conserved.
Before collision:
- Total momentum: $mu + 0 = mu$
- Total kinetic energy: $\frac{1}{2}mu^2 + 0 = \frac{1}{2}mu^2$
After collision:
- Total momentum: $0 + mu = mu$
- Total kinetic energy: $0 + \frac{1}{2}mu^2 = \frac{1}{2}mu^2$
Example 2: Collision with Different Masses and Velocities
Let's consider two objects with masses $m_1$ and $m_2$, and velocities $u_1$ and $u_2$ respectively. After an elastic collision, their velocities change to $v_1$ and $v_2$. Using the conservation laws, we can find the final velocities:
Conservation of momentum:
$$ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 $$
Conservation of kinetic energy:
$$ \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 $$
Solving these equations simultaneously gives us the final velocities $v_1$ and $v_2$.
Example 3: Head-on Collision of Two Objects
For a head-on elastic collision of two objects with masses $m_1$ and $m_2$, and initial velocities $u_1$ and $u_2$, the final velocities can be derived as:
$$ v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2} $$
$$ v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2} $$
These equations show how the final velocities depend on both the masses and the initial velocities of the objects involved in the collision.
In conclusion, understanding elastic collisions is crucial for solving problems in physics where energy and momentum conservation are involved. The ability to apply the conservation laws to calculate the final states of objects after a collision is a fundamental skill in physics.