Inelastic Collisions

The conservation of kinetic energy law is not applicable in inelastic collisions. The conservation of momentum law is applicable in inelastic collisions if only no external force acts on the two colliding objects. In an inelastic collision, two objects stick together or are attached to each other after the collision.

Example question 1.

Two objects are of the same mass, namely 1 kg. Object 1 moves on a flat plane at a speed of 10 m/s and collides with object two which is at rest. After the collision, the two objects stick together. What is the speed of the two objects after the collision?

__Known :__

m_{1 }= 1 kg, m_{2 }= 1 kg, v_{1 }= 10 m/s, v_{2} = 0

__Wanted__ : v’

__Solution :__

m_{1} v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) v’

(1 kg)(10 m/s) + 0 = (1 kg + 1 kg) v’

10 kg m/s = (2 kg) v’

v’ = 10 kg m/s : 2 kg = 5 m/s

Example question 2.

Three blocks move at 3 m.s^{-1} collide another block at rest.

The collision is inelastic. The order of block’s velocity after the collision, from the largest to the smallest is…

Solution :

Figure 1 :

Final momentum = initial momentum

m_{1} v_{1} + m_{2} v_{2 }= (m_{1 }+ m_{2}) v

(4m)(3) + (m)(0) = (4m + m) v

12m + 0 = (5m) v

12m = 5m v

v = 12m / 5m = 12/5 = 2.4 m/s

Figure 2 :

Final momentum = Initial momentum

m_{1} v_{1} + m_{2} v_{2 }= (m_{1 }+ m_{2}) v

(m)(3) + (3m)(0) = (m + 3m) v

3m + 0 = (4m) v

3m = 4m v

v = 3m / 4m = 3/4 = 0.75 m/s

Figure 3 :

m_{1} v_{1} + m_{2} v_{2 }= (m_{1 }+ m_{2}) v

(m)(3) + (m)(0) = (m + m) v

3m + 0 = (2m) v

3m = 2m v

v = 3m / 2m = 3/2 = 1.5 m/s

Example question 3.

Two balls with mass of m_{1} = 2 kg and m_{2} = 1 kg are move in opposite direction with speed of v_{1} = 2 ms^{-1} and v_{2} = 4 ms^{-1} as shown in figure below. If a collision is inelastic, what is the speed of both balls after the collision?

__Known :__

Mass of ball 1 (m_{1}) = 2 kg

Mass of ball 2 (m_{2}) = 1 kg

Velocity of ball 1 before collision (v_{1}) = 2 m/s

Velocity of ball 2 before collision (v_{2}) = -4 m/s

Plus and minus sign indicates that both balls move in opposite direction.

__Wanted ____:__ Velocity of balls after collision (v’)

__Solution :__

m_{1} v_{1} + m_{2} v_{2} = (m_{1} + m_{2}) v’

(2)(2) + (1)(-4) = (2 + 1) v’

4 – 4 = (3) v’

0 = (3) v’

v’ = 0

Example question 4.

Two objects, A and B, with a mass of each object, is 1.5-kg approach each other with speed of v_{A} = 4 m.s^{-1} and v_{B} = 5 m.s^{-1}. If the collision is inelastic, what is the speed of both objects after the collision?

__Known :__

Mass of object A (m_{A}) = 1.5 kg

Mass of object B (m_{B}) = 1.5 kg

Velocity of object A before collision (v_{A}) = 4 m/s (plus sign, to rightward)

Velocity of object B before collision (v_{B}) = -5 m/s (minus sign, to leftward)

__Wanted :__ The speed of both objects after collision

__Solution :__

Conservation of linear momentum :

m_{A }v_{A} + m_{B }v_{B} = (m_{A} + m_{B}) v’

(1.5)(4) + (1.5)(-5) = (1.5 +1.5) v’

6 – 7.5 = (3) v’

-1.5 = (3) v’

v’ = -1.5 / 3

v’ = -0.5 m/s

Minus sign indicates that both objects move to leftward.

20 conceptual questions and answers about inelastic collisions:

**1. Question:** What defines an inelastic collision? **Answer:** In an inelastic collision, kinetic energy is not conserved, though momentum is. Some of the initial kinetic energy is transformed into other forms of energy.

**2. Question:** How does a perfectly inelastic collision differ from a partially inelastic collision? **Answer:** In a perfectly inelastic collision, the objects stick together after the collision. In a partially inelastic collision, the objects separate, but there’s still a loss of kinetic energy.

**3. Question:** What remains conserved in all types of collisions, including inelastic ones? **Answer:** Momentum is always conserved in all collisions, regardless of their elasticity.

**4. Question:** Why isn’t kinetic energy conserved in inelastic collisions? **Answer:** Some of the kinetic energy gets converted into other forms of energy, such as potential energy, heat, or sound.

**5. Question:** How can one identify an inelastic collision just by observing the velocities before and after the collision? **Answer:** The total kinetic energy before the collision will be greater than the total kinetic energy after the collision.

**6. Question:** Can inelastic collisions occur in one dimension only? **Answer:** No, inelastic collisions can occur in one, two, or three dimensions. The principles remain the same, only the vector calculations become more complex.

**7. Question:** In the context of particle physics, what’s a common outcome of inelastic collisions? **Answer:** In particle physics, inelastic collisions often result in the transformation of the colliding particles into different particles.

**8. Question:** How can you determine the amount of energy lost in an inelastic collision? **Answer:** By calculating the difference between the total initial kinetic energy and the total final kinetic energy.

**9. Question:** Why don’t inelastic collisions violate the law of conservation of energy? **Answer:** Energy is still conserved; it’s just converted from one form (kinetic) to others (like heat or sound) rather than remaining purely as kinetic energy.

**10. Question:** Are most real-world collisions inelastic? **Answer:** Yes, most real-world collisions are inelastic because there’s typically some conversion of kinetic energy to other forms.

**11. Question:** In an inelastic collision, if two objects stick together, what can be said about their combined velocity? **Answer:** Their combined velocity is determined by the conservation of momentum. The two objects will move with a common velocity after the collision.

**12. Question:** How does the coefficient of restitution relate to inelastic collisions? **Answer:** The coefficient of restitution, denoted as $e$, measures the “bounciness” of a collision. For perfectly inelastic collisions, $e=0$.

**13. Question:** Why might an inelastic collision produce sound? **Answer:** The collision can cause vibrations in the colliding objects, which may produce sound waves in the surrounding medium.

**14. Question:** Can gravitational potential energy be a factor in inelastic collisions? **Answer:** Yes, especially if the collision results in a change in height or position of the objects, converting kinetic energy into gravitational potential energy.

**15. Question:** Why do rubber balls not undergo perfectly inelastic collisions, despite being “bouncy”? **Answer:** Rubber balls undergo elastic or near-elastic collisions because they tend to retain much of their kinetic energy and bounce back. They don’t stick together, which would be characteristic of a perfectly inelastic collision.

**16. Question:** Can two soft clay balls demonstrate a perfectly inelastic collision? **Answer:** Yes, because when two soft clay balls collide, they tend to stick together and not bounce back, which is characteristic of a perfectly inelastic collision.

**17. Question:** Is it possible for an inelastic collision to occur without any sound or heat generation? **Answer:** It’s rare, but possible. The energy could be dissipated in other subtle ways or stored as internal potential energy.

**18. Question:** Can inelastic collisions be reversed? **Answer:** Generally, inelastic collisions are not reversible because the conversion of kinetic energy to other forms makes it difficult to revert to the initial state.

**19. Question:** How does air resistance relate to inelastic collisions? **Answer:** Air resistance can make collisions more inelastic by dissipating some of the kinetic energy as heat.

**20. Question:** Do inelastic collisions have implications in safety designs, such as car crumple zones? **Answer:** Yes, crumple zones in cars are designed to undergo inelastic collisions, absorbing kinetic energy and reducing the forces acting on occupants.

Inelastic collisions play a vital role in understanding the conservation laws of physics and are integral in numerous practical applications and safety designs.