What is Elastic Collision Overview
When two objects collide, their kinetic energy is transferred between them, resulting in changes in their motion. An elastic collision is a type of collision in which there is no net loss of kinetic energy, and both momentum and energy are conserved. In this article, we will discuss the basics of elastic collisions, their characteristics, equations, and real-world applications.
What is Elastic Collision?
An elastic collision is a type of collision in which the kinetic energy of the system is conserved. This means that the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision. In elastic collisions, the colliding objects bounce off each other without any deformation or loss of energy.
What is Elastic Collisions Characteristics
Elastic collisions have the following characteristics:
- Conservation of Kinetic Energy: The kinetic energy of the system is conserved in elastic collisions, which means that the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision.
- Conservation of Momentum: Momentum is also conserved in elastic collisions, which means that the total momentum of the system before the collision is equal to the total momentum after the collision.
- No Loss of Energy: Elastic collisions do not result in any loss of kinetic energy due to deformation or friction. Therefore, the colliding objects bounce off each other without losing energy.
- No External Forces: Elastic collisions occur in a closed system without any external forces acting on the system.
What is Elastic Collision Formula
The Elastic Collision formula of momentum is given by:
m1u1 + m2u2 = m1v1 + m2v2
Where,
- m1 = Mass of 1st body
- m2 = Mass of 2nd body
- u1 =Initial velocity of 1st body
- u2 = Initial velocity of the second body
- v1 = Final velocity of the first body
- v2 = Final velocity of the second body
The Elastic Collision formula of kinetic energy is given by:
(1/2) m1u12 + (1/2) m2u22 = (1/2) m1v12 + (1/2) m2v22
What is Elastic Collisions Example
One common example of an elastic collision is a game of billiards. When the cue ball strikes another ball, the collision between the two is almost perfectly elastic. This means that the kinetic energy of the cue ball is transferred to the second ball without any loss of energy. The second ball then moves away from the cue ball with the same speed and direction as the cue ball had before the collision.
Another example of an elastic collision is two molecules colliding in a gas. In this case, the kinetic energy and momentum of the two molecules are conserved, and they bounce off each other without any loss of energy. Elastic collisions between gas molecules are responsible for the transfer of heat and energy in a gas.
In both of these examples, the colliding objects bounce off each other without any deformation or loss of energy. This is a characteristic of elastic collisions, and it demonstrates the conservation of kinetic energy and momentum in these types of collisions.
Elastic Collision Example Equations
Two billiard balls collide. Ball 1 moves with a velocity of 6 m/s, and ball 2 is at rest. After the collision, ball 1 comes to a complete stop. What is the velocity of ball 2 after the collision? Is this collision elastic or inelastic? The mass of each ball is 0.20 kg.
Solution:
To find the velocity of ball 2, use a momentum table.
|
Objects |
Momentum Before |
Momentum After |
|
Ball 1 |
0.20 kg × 6 m/s = 1.2 |
0 |
|
Ball 2 |
0 |
0.20 kg × v2 |
|
Total |
1.2 kg × m/s |
0.20 kg × v2 |
1.2 kg × m/s = 0.20 kg × v2
v2 =1.2 / 0.20 = 6 m/s
To determine whether the collision is elastic or inelastic, calculate the total kinetic energy of the system both before and after the collision.
|
Objects |
KE Before (J) |
KE After (J) |
|
Ball 1 |
0.50 × 0.20 × 62 = 3.6 |
0 |
|
Ball 2 |
0 |
0.50 × 0.20 × 62 = 3.6 |
|
Total |
3.6 |
3.6 |
Since the kinetic energy before the collision equals the kinetic energy after the collision (kinetic energy is conserved), this is an elastic collision.
The Mathematical Formulation of Elastic Collision
The conservation of kinetic energy principle can be expressed mathematically as follows:
m1v1i + m2v2i = m1v1f + m2v2f
Where,
m1 and m2 are the masses of the colliding objects, v1i and v2i are their initial velocities, and v1f and v2f are their final velocities. This equation states that the sum of the initial kinetic energies of the two objects (m1v1i^2/2 + m2v2i^2/2) is equal to the sum of their final kinetic energies (m1v1f^2/2 + m2v2f^2/2).
The principle of conservation of momentum is also applicable to elastic collisions. According to this principle, the total momentum of the system before and after the collision is also conserved. Mathematically, this can be expressed as:
m1v1i + m2v2i = m1v1f + m2v2f
Where,
p1i = m1v1i and p2i = m2v2i are the initial momenta of the two objects, and p1f = m1v1f and p2f = m2v2f are their final momenta.
By combining the conservation of kinetic energy and the conservation of momentum principles, we can solve for the final velocities of the objects after the collision. This leads to the following equations:
v1f = (m1 - m2)/(m1 + m2) * v1i + (2m2)/(m1 + m2) * v2i
v2f = (m2 - m1)/(m1 + m2) * v2i + (2m1)/(m1 + m2) * v1i
These equations show how the final velocities of the two objects depend on their initial velocities and masses.
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Difference between Elastic and Inelastic Collision
|
Elastic Collision |
Inelastic Collision |
|
The total kinetic energy is conserved. |
The total kinetic energy of the bodies at the beginning and the end of the collision is different. |
|
Momentum is conserved. |
Momentum is conserved. |
|
No conversion of energy takes place. |
Kinetic energy is changed into other energy such as sound or heat energy. |
|
Highly unlikely in the real world as there is almost always a change in energy. |
This is the normal form of collision in the real world. |
|
An example of this can be swinging balls or a spacecraft flying near a planet but not getting affected by its gravity in the end. |
An example of an inelastic collision can be the collision of two cars. |
What is Elastic Collision Applications
- The collision time affects the amount of force an object experiences during a collision. The greater the collision time, the smaller the force acting upon the object. Thus, to maximize the force experienced by an object during a collision, the collision time must be decreased.
- Likewise, the collision time must be increased to minimize the force. There are several real-world applications of these phenomena. The airbags in automobiles increase the collapse time and minimize the effect of force on objects during a collision. The airbag accomplishes this by extending the time required to stop the momentum of the passenger and the driver.
Things to Remember about Elastic Collision
- Elastic collision is a type of collision between two objects in which kinetic energy is conserved.
- In an elastic collision, both the momentum and the kinetic energy of the objects are conserved.
- Elastic collisions can only occur between objects that are elastic, meaning they can return to their original shape after being deformed.
- Inelastic collisions, on the other hand, are collisions in which some of the kinetic energy is lost, usually due to deformation or friction.
- In an elastic collision, the total momentum before and after the collision remains the same, while in an inelastic collision, the total momentum may change.
- Elastic collisions are idealized and do not occur in real-life situations, but they are useful for understanding the behavior of objects in a collision.
- The coefficient of restitution (e) is used to describe the elasticity of a collision. It is defined as the ratio of the relative velocities of the objects after and before the collision.
- The coefficient of restitution ranges from 0 to 1, with 0 indicating a completely inelastic collision and 1 indicating a completely elastic collision.
- Elastic collisions are important in fields such as physics and engineering, where they are used to model and analyze the behavior of objects in motion.
- In elastic collisions, the angle of incidence is equal to the angle of reflection, as described by the law of reflection. This is true for both one-dimensional and two-dimensional collisions.