Partially elastic collisions

In partially elastic collisions, the law of conservation of momentum is applicable, while the conservation of kinetic energy law is not applicable. At the time a collision takes place, some kinetic energy is converted to sound energy, heat energy, and internal energy. The use of the word elastic signifies that after the collision, the two objects do not stick together but bounce off.

An example of partially elastic collision is the one-dimensional collision of two marbles or two pool balls.

Example question 1.

Objects A and B with masses of 1 kg and 2 kg, respectively move in opposite directions at speeds of 4 m/s and 2 m/s, respectively and collide in a partially elastic collision. If after collision object A moves at a speed of 2 m/s, what is the speed of object B?

Known :

m_{A} = 1 kg, m_{B} = 2 kg, v_{A} = 4 m/s (suppose to the right), v_{B} = -2 m/s (suppose to the left), v_{A }’ = – 2 m/s

Wanted : v_{B} ’

Solution :

m_{A} v_{A} + m_{B }v_{B }= m_{A} v_{A} ’ + m_{B} v_{B} ’

(1 kg)(4 m/s) + (2 kg)(-2 m/s) = (1 kg)(-2 m/s) + (2 kg)(v_{B}’)

4 kg m/s – 4 kg m/s = -2 kg m/s + (2 kg) v_{B}’

0 = – 2 kg m/s + (2 kg)(v_{B}’)

-2 kg m/s = – 2 kg v_{B }’

v_{B}’ = 1 m/s

After collision, object B moves at a speed of 1 m/s (suppose to the right) and object A moves at a speed of 2 m/s (suppose to the left)

Example question 2.

Object A and B with a mass of 4-kg and 5-kg approach each other in the opposite direction, as shown in the figure below. After collision, both objects reversed direction with speed of A = 4 m.s^{-1} and speed of B = 2 m.s^{–}^{1}. What is the speed of object B before the collision?

__Known :__

Mass of object A (m_{A}) = 4 kg

Mass of object B (m_{B}) = 5 kg

Velocity of object A before collision (v_{A}) = 6 m/s

Velocity of object A after collision (v_{A}’) = 4 m/s

Velocity of object B after collision (v_{B}’) = -2 m/s

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

__Wanted :__ Velocity of object B before collision (v_{B})

__Solution :__

m_{A} v_{A} + m_{B} v_{B} = m_{A} v_{A}’ + m_{B} v_{B}’

(4)(6) + (5)(-2) = (4)(4) + (5)(v_{B}’)

24 – 10 = 16 + 5(v_{B}’)

14 – 16 = 5 (v_{B}’)

-2 = 5 (v_{B}’)

v_{B}’ = -2/5

v_{B}’ = -0.4

Minus sign indicates that object direction after collision is different with direction before collision.

Example question 3.

Two objects with the same mass approach each other as shown in the figure below.

If v_{2}‘ is the speed of the object (2) after the collision to rightward with speed of 5 m.s^{–1}, what is the speed of object one v_{1 }‘ (1) after the collision?

__Known :__

Mass of each object = m

Velocity of object 1 before collision (v_{1}) = 8 m/s

Velocity of object 2 before collision (v_{2}) = 10 m/s

Velocity of object 2 after collision (v_{2}‘) = 5 m/s

__Wanted :__ Speed of object 1 after collision (v_{1}‘)

__Solution :__

m_{1} v_{1}+ m_{2 }v_{2} = m_{1 }v_{1}’ + m_{2 }v_{2}’

m (v_{1 }+ v_{2}) = m (v_{1}’ + v_{2}’)

v_{1} + v_{2} = v_{1}’ + v_{2}’

8 + 10 = v_{1}’ + 5

18 = v_{1}’ + 5

v_{1}’ = 18-5

v_{1}’ = 13 m/s

20 conceptual questions and answers about partially elastic collisions:

**1. Question:** What is a partially elastic collision? **Answer:** A partially elastic collision is one where some kinetic energy is conserved, but not all. The objects rebound off each other but not with their full initial kinetic energies.

**2. Question:** How does a partially elastic collision differ from a perfectly elastic collision? **Answer:** In a perfectly elastic collision, the total kinetic energy is conserved, while in a partially elastic collision, only a portion of the kinetic energy is conserved.

**3. Question:** What remains conserved in a partially elastic collision? **Answer:** Momentum remains conserved in all types of collisions, including partially elastic ones.

**4. Question:** How is the coefficient of restitution, $e$, used to describe the elasticity of a collision? **Answer:** For partially elastic collisions, $0<e<1$. The closer $e$ is to 1, the more elastic the collision.

**5. Question:** Can the coefficient of restitution be greater than 1? **Answer:** No. A coefficient of restitution greater than 1 would imply more kinetic energy after the collision than before, which violates energy conservation.

**6. Question:** Why isn’t all kinetic energy conserved in partially elastic collisions? **Answer:** Some kinetic energy is transformed into other forms, such as sound, heat, or deformation of the objects.

**7. Question:** How can you determine the amount of kinetic energy retained in a partially elastic collision? **Answer:** By comparing the total kinetic energy before and after the collision.

**8. Question:** Do partially elastic collisions always produce sound? **Answer:** Not always, but sound can be produced due to vibrations caused by the collision.

**9. Question:** What happens to the kinetic energy that isn’t conserved in a partially elastic collision? **Answer:** It’s typically transformed into other forms of energy like heat, sound, or potential energy due to deformation.

**10. Question:** How does material composition affect the elasticity of a collision? **Answer:** Some materials, like rubber, tend to undergo more elastic collisions, while others, like clay, tend to be inelastic. However, most real-world collisions are partially elastic due to the materials’ properties and other factors.

**11. Question:** Can two steel balls demonstrate a partially elastic collision? **Answer:** Yes. While steel balls may seem to have a nearly perfectly elastic collision, there’s usually a slight energy loss, making it partially elastic.

**12. Question:** Is the Earth’s collision with a meteorite considered partially elastic? **Answer:** Generally, yes. The impact doesn’t conserve all kinetic energy as some is transformed into heat, sound, and the creation of a crater.

**13. Question:** Can you have a partially elastic collision in a vacuum? **Answer:** Yes. Even without air resistance, the internal properties of the colliding bodies can lead to energy losses.

**14. Question:** How do external forces, like friction, affect the elasticity of a collision? **Answer:** External forces can dissipate more kinetic energy, making the collision more inelastic.

**15. Question:** Are there practical applications where understanding partially elastic collisions is crucial? **Answer:** Yes, in sports, engineering, and traffic accident analysis, understanding partially elastic collisions can be essential.

**16. Question:** Why do car manufacturers study partially elastic collisions? **Answer:** To design vehicles that can absorb impact energy efficiently, protecting the occupants during collisions.

**17. Question:** Can particle interactions at the quantum level be considered partially elastic? **Answer:** Yes. While quantum mechanics introduces new phenomena, particles can still undergo interactions where not all kinetic energy is conserved.

**18. Question:** How do molecular interactions relate to partially elastic collisions? **Answer:** Molecules can undergo partially elastic collisions when they hit each other, transferring some energy into vibrations or rotations of the molecules.

**19. Question:** Does a basketball bouncing on a court demonstrate a partially elastic collision? **Answer:** Yes. Some energy is lost as sound and to the slight deformation of the ball, making the collision partially elastic.

**20. Question:** How does temperature affect the elasticity of collisions in gases? **Answer:** As temperature increases, gas molecules move faster and can collide more energetically. However, the elasticity of these collisions depends on molecular properties and doesn’t directly correlate with temperature.

Understanding partially elastic collisions provides insights into a range of phenomena in both everyday experiences and specialized scenarios in physics and engineering.