Types of Collisions: Elastic, Inelastic, and Perfectly Inelastic
Collisions
Collisions are interactions where two or more objects exert forces on each other for a short time.
Momentum is always conserved in collisions, but kinetic energy may or may not be conserved.
Elastic Collisions
Elastic collision
In an elastic collision, both momentum and kinetic energy are conserved.
- Imagine two billiard balls colliding.
- After the collision, they move apart with the same total kinetic energy they had before.
Mathematically, for two objects with masses $m_1$ and $m_2$ and initial velocities $u_1$ and $u_2$:
- Momentum conservation: $$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$
- Kinetic energy conservation: $$\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$$
Inelastic Collisions
Inelastic collision
In an inelastic collision, momentum is conserved, but kinetic energy is not.
Some kinetic energy is transformed into other forms, such as heat or sound.Example
When a car crashes into a barrier, the car deforms, and energy is lost as heat and sound.
Perfectly Inelastic Collisions
Perfectly inelastic collision
In a perfectly inelastic collision, the colliding objects stick together and move as one mass after the collision.
This type of collision results in the maximum possible loss of kinetic energy.Example
Two clay balls collide and merge into a single mass, moving together with a common velocity.
Elastic Collision
Consider two blocks colliding elastically:
- Block 1: mass = 2 kg, velocity = 5 $\text{m s}^{-1}$
- Block 2: mass = 3 kg, velocity = 0 $\text{m s}^{-1}$
Calculate their velocities after collision.
Solution
Before Collision:
Total momentum:
$$p_{\text{initial}} = (2)(5) + (3)(0) = 10 \, \text{kg m s}^{-1}$$
After Collision:
Using momentum conservation:
$$2u_1 + 3u_2 = 10 \quad \text{(1)}$$
Using energy conservation:
$$\frac{1}{2}(2)(5^2) = \frac{1}{2}(2)u_1^2 + \frac{1}{2}(3)$$
$$25 = u_1^2 + \frac{3}{2}u_2^2 \quad \text{(2)}$$
Solving Equations:
From (1):
$$u_1 = 5 - \frac{3}{2}u_2 \quad \text{(3)} $$
Substitute (3) into (2):
$$25 = (5 - \frac{3}{2}u_2)^2 + \frac{3}{2}u_2^2$$
Solve to get:
$$u_1 = -1 \, \text{m s}^{-1}, \quad u_2 = 4 \, \text{m s}^{-1}$$
Inelastic Collision
Consider two blocks colliding and sticking together:
- Block 1: mass = 4 kg, velocity = 6 $\text{m s}^{-1}$
- Block 2: mass = 8 kg, velocity = 0 $\text{m s}^{-1}$
Calculate their common velocity after collision.
Solution
Before Collision:
$$\text{Total momentum} = (4 \text{ kg} × 6 \text{ m s}^{-1}) + (8 \text{ kg} × 0 \text{ m s}^{-1}s) = 24 \text{ kg m s}^{-1}$$
After Collision:
The blocks move together with a common velocity $v$:
$$\text{Total momentum} = (4 text{ kg} + 8 text{ kg}) × v = 12v$$
Setting the total momentum before and after the collision equal:
$$
12v = 24 \implies v = 2 \, \text{m s^}{-1}
$$
Dissipation of Energy in Collisions:
Kinetic Energy Dissipation
In inelastic and perfectly inelastic collisions, some kinetic energy is transformed into other forms, such as:
- Thermal energy: Heat generated by friction or deformation.
- Sound energy: Noise produced during the collision.
- Deformation energy: Energy used to permanently deform the objects.
The law of conservation of energy still holds, but kinetic energy is not conserved because it is converted into other forms.
