Collisions
Collisions are interactions where two or more objects exert forces on each other for a short time.
Momentum is conserved in collisions if external forces on the system are negligible, but kinetic energy may or may not be conserved.
Elastic collision
In an elastic collision, both momentum and kinetic energy are conserved.
Mathematically, for two objects with masses $m_1$ and $m_2$ and initial velocities $u_1$ and $u_2$:
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.
When a car crashes into a barrier, the car deforms, and energy is lost as heat and sound.
Perfectly inelastic collision
In a perfectly inelastic collision, the colliding objects stick together and move as one mass after the collision.
In a perfectly inelastic collision, the objects stick together and move with a common final velocity, and the kinetic energy loss depends on the masses and initial velocities.
Two clay balls collide and merge into a single mass, moving together with a common velocity.
Consider two blocks colliding elastically:
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:
$$2v_1 + 3v_2 = 10 \quad \text{(1)}$$
Using energy conservation:
Using energy conservation: $$\frac{1}{2}(2)(5^2) + \frac{1}{2}(3)(0^2) = \frac{1}{2}(2)v_1^2 + \frac{1}{2}(3)v_2^2$$
$$25 = v_1^2 + \frac{3}{2}v_2^2 \quad \text{(2)}$$
Solving Equations:
From (1):
$$v_1 = 5 - \frac{3}{2}v_2 \quad \text{(3)}$$
Substitute (3) into (2):
$$25 = \left(5 - \frac{3}{2}v_2\right)^2 + \frac{3}{2}v_2^2$$
Solve to get:
$$v_1 = -1 \, \text{m s}^{-1}, \quad v_2 = 4 \, \text{m s}^{-1}$$
Consider two blocks colliding and sticking together:
Calculate their common velocity after collision.
Solution
Before Collision:
$$\text{Total momentum} = (4\text{ kg} \times 6\text{ m s}^{-1}) + (8\text{ kg} \times 0\text{ m s}^{-1}) = 24\text{ kg m s}^{-1}$$
After Collision:
The blocks move together with a common velocity $v$:
$$\text{Total momentum} = (4\text{ kg} + 8\text{ kg}) \times v = 12v$$
Setting the total momentum before and after the collision equal:
$$12v = 24 \implies v = 2 \, \text{m s}^{-1}$$
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