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Elastic Collision Formula - Definition, Formula, Solved Problems

Elastic Collision Formula: Elastic collisions are a fundamental concept in physics, and they find applications in various fields.
authorImageMurtaza Mushtaq12 Oct, 2023
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Elastic Collision Formula

Elastic Collision Formula: An elastic collision is a type of collision between two objects in which both kinetic energy and momentum are conserved. In other words, when two objects collide elastically, they rebound off each other without any loss of kinetic energy. This is in contrast to inelastic collisions where kinetic energy is not conserved.

Elastic Collision Formula

To analyze and calculate elastic collisions, we use the following equations:
  1. Conservation of Momentum: The total momentum before the collision equals the total momentum after the collision.
m1 * u1 _initial + m2 * u2_i nitial = m1 * v1_ final + m2 * v2 _final - m1 and m2 represent the masses of the colliding objects. - u1_initial and u2_initial are the initial velocities of the objects. - v1_final and v2_final represent the final velocities of the objects.
  1. Conservation of Kinetic Energy: The total kinetic energy before the collision equals the total kinetic energy after the collision.
0.5 * m1 * (u1_ initial ) 2 + 0.5 * m2 * (u2_ initial ) 2 = 0.5 * m1 * (v1 _final ) 2 + 0.5 * m2 * (v2_ final ) 2 These equations are instrumental in determining the final velocities of objects involved in elastic collisions.

Also Check - Heat Input Formula

Solved Problems Of Elastic Collision Formula

An elastic collision is a type of collision between two objects in which both kinetic energy and momentum are conserved. In other words, when two objects collide elastically, they rebound off each other without any loss of kinetic energy. This is in contrast to inelastic collisions where kinetic energy is not conserved. Problem 1: Billiard Balls Two billiard balls, one with a mass of 0.2 kg and an initial velocity of 2 m/s, and the other with a mass of 0.3 kg and an initial velocity of -1 m/s, collide elastically. Calculate their final velocities. To solve this, we'll use the equations for conservation of momentum and kinetic energy:

Conservation of Momentum:

m1 * u1_initial + m2 * u2_initial = m1 * v1_final + m2 * v2_final

Conservation of Kinetic Energy:

0.5 * m1 * (u1_initial)^2 + 0.5 * m2 * (u2_initial)^2 = 0.5 * m1 * (v1_final)^2 + 0.5 * m2 * (v2_final)^2

Also Check - Critical Velocity Formula

Given:

- m1 = 0.2 kg - u1_initial = 2 m/s - m2 = 0.3 kg - u2_initial = -1 m/s We need to find v1_final and v2_final Using these equations, you can calculate the final velocities of the two billiard balls.

Also Check - Heat Of Vaporization Formula

Problem 2: Elastic Collision

Two cars, Car A and Car B, collide head-on. Car A has a mass of 1000 kg and is initially moving to the right at 20 m/s, while Car B has a mass of 800 kg and is initially moving to the left at 15 m/s. The collision between the two cars is perfectly elastic. Calculate the final velocities of Car A and Car B after the collision.

Solution:

In an elastic collision, both momentum and kinetic energy are conserved. To find the final velocities of the two cars, we can use these principles. Step 1: Conservation of Momentum The total momentum before the collision is equal to the total momentum after the collision. Initial momentum of Car A = 1000 kg * 20 m/s = 20000 kg·m/s (to the right) Initial momentum of Car B = 800 kg * (-15 m/s) = -12000 kg·m/s (to the left) The total initial momentum = 20000 kg·m/s (to the right) - 12000 kg·m/s (to the left) = 8000 kg·m/s (to the right). The total momentum after the collision should also be 8000 kg·m/s to the right, as momentum is conserved.

Also Check - Strain Formula

Step 2: Conservation of Kinetic Energy The total kinetic energy before the collision is equal to the total kinetic energy after the collision. Initial kinetic energy of Car A = 0.5 * 1000 kg * (20 m/s)^2 = 200,000 J Initial kinetic energy of Car B = 0.5 * 800 kg * (15 m/s)^2 = 90,000 J The total initial kinetic energy = 200,000 J + 90,000 J = 290,000 J. The total kinetic energy after the collision should also be 290,000 J, as kinetic energy is conserved. Now, we have the initial momentum and kinetic energy, and we know they are conserved after the collision. We can calculate the final velocities using the conserved values of momentum and kinetic energy. The final velocity of Car A = Total momentum / Mass of Car A Final velocity of Car A = 8000 kg·m/s / 1000 kg = 8 m/s to the right. The final velocity of Car B = Total momentum / Mass of Car B Final velocity of Car B = 8000 kg·m/s / 800 kg = 10 m/s to the right. So, the final velocity of Car A is 8 m/s to the right, and the final velocity of Car B is 10 m/s to the right. Both cars change direction and have their speeds adjusted in the direction of Car B's initial motion. For this problem, you can apply the same conservation of momentum and kinetic energy equations, but you have the advantage of knowing that the two balls have equal mass. Given that m1 = m2, and u1_initial = -u2_initial, you can simplify the equations and calculate their final velocities.

Applications of Elastic Collision Formula

Elastic collisions have numerous applications in the real world. They are used in:
  1. Car Safety: Understanding elastic collisions is essential for designing safety features in vehicles and crash testing.
  2. Ballistics: The study of the motion of projectiles, including bullets, relies on the principles of elastic collisions.
  3. Particle Physics: In particle accelerators, particles undergo elastic collisions, allowing scientists to study subatomic particles.
  4. Astronomy : Elastic collisions play a role in understanding the motion of celestial bodies and their interactions in space.
  5. Sports : The behavior of balls in various sports, such as billiards, snooker, and pool, can be described using elastic collision principles.
In conclusion, elastic collisions are a fundamental concept in physics, and understanding the elastic collision formula is essential for solving problems and analyzing real-world situations. Their applications range from car safety to particle physics, making them a valuable topic of study for scientists and engineers.

Elastic Collision Formula FAQs

What is the primary difference between elastic and inelastic collisions?

 In elastic collisions, both kinetic energy and momentum are conserved, while in inelastic collisions, only momentum is conserved.

Why are elastic collisions important in physics and engineering?

Elastic collisions provide insights into how objects interact while conserving energy, making them crucial in various fields.

Can you provide more examples of elastic collisions in daily life?

 Certainly, you can observe elastic collisions in scenarios like air molecules in a gas, the rebounding of basketballs, and the dynamics of atoms and particles.

Is it possible to achieve perfectly elastic collisions in the real world?

Perfectly elastic collisions are idealized and rarely achieved due to external factors like air resistance, but they can be closely approximated.
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