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 Collisions

Collisions are fascinating events where a strong force acts between two or more bodies for a very short time, leading to changes in their energy and momentum.

 Whether it's a collision between billiard balls or an interaction on a subatomic level, collisions showcase the key principles of physics.

1. Stages of Collision: 

Collisions can be broken into three stages—before, during, and after the interaction.

  • Before and after the collision, forces are absent. 
  • During the collision, large forces dominate and govern the motion.

Collisions | Physics Class 11 - NEET

2. Conservation of Momentum and Energy: 

  • In an ideal system, the total momentum and energy remain conserved throughout the collision. 
  • This principle helps us relate initial and final velocities of the colliding objects, providing a deep insight into how motion evolves.

Types of Collisions

(i) On the basis of conservation of kinetic energy.

Collisions | Physics Class 11 - NEET

(ii) On the basis of the direction of colliding bodies:

Collisions | Physics Class 11 - NEET

Question for Collisions
Try yourself:
Which type of collision is based on the conservation of kinetic energy?
View Solution

1. Elastic collision

The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions.

  • In an elastic collision all the involved forces are conservative forces.
  • Total energy remains conserved.

2. Inelastic collision

The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions.

  • In an inelastic collision some or all the involved forces are non-conservative forces.
  • Total energy of the system remains conserved.
  • If after the collision two bodies stick to each other, then the collision is said to be perfectly inelastic.

One-Dimensional or Head-on Collisions

If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or head-on collision.

Inelastic One Dimensional Collision

Applying Newton’s experimental law, we have

Collisions | Physics Class 11 - NEET

  • Velocities after collision:
    v1 = (m1 – m2) u1 + 2m2u2 / (m1 + m2)
     v2 = (m2 – m1) u2 + 2m1u1 / (m1 + m2)
  • When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities.
    v1 = u2 and v2 = u1
  • If the second body of same mass (m1 = m2) is at rest, then after collision first body comes to rest and second body starts moving with the initial velocity of the first body.
    v1 = 0 and v2 = u1
  • If a light body of mass m1 collides with a very heavy body of mass m2 at rest, then after a collision.
    v1 = – u1 and v2 = 0
    It means the light body will be rebound with its own velocity and heavy body will continue to be at rest.
  • If a very heavy body of mass m1 collides with a light body of mass m2(m1 > > m2) at rest, then after collision
    v1 = u1 and v2 = 2u1
  • Loss of kinetic energy:
    ΔE = m1m2 / 2(m1 + m2) (u1 – u2)2 (1 – e2)

Example 1:

Collisions | Physics Class 11 - NEET

Sol: (c) 1:3

Explanation:

Collisions | Physics Class 11 - NEET

 Perfectly Inelastic One Dimensional Collision

In such types of collisions, the bodies move independently before the collision but after the collision, they move as one single body.

Collisions | Physics Class 11 - NEET

(i) When the colliding bodies are moving in the same direction:

Collisions | Physics Class 11 - NEET

Loss in kinetic energy:

Collisions | Physics Class 11 - NEET

(ii) When the colliding bodies are moving in the opposite direction:

Before CollisionBefore Collision

Collisions | Physics Class 11 - NEET

Loss in kinetic energy:

Collisions | Physics Class 11 - NEET

Example 2: A body of mass 2 kg is placed on a horizontal frictionless surface. It is connected to one end of a spring whose force constant is 250 N/m. The other end of the spring is joined with the wall. A particle of mass 0.15 kg moving horizontally with speed v sticks to the body after collision. If it compresses the spring by 10 cm, the velocity of the particle is

(a) 3 m/s

(b) 5 m/s

(c) 10 m/s

(d) 15 m/s

Sol:

 Collisions | Physics Class 11 - NEET

Collisions | Physics Class 11 - NEET

Perfectly Elastic Head on Collision

Let two bodies of masses m1 and mmoving with initial velocities u1 and u2  in the same direction and they collide such that after collision their final velocities are v1 and v2  respectively.

Collisions | Physics Class 11 - NEET

According to the law of conservation of momentum:

Collisions | Physics Class 11 - NEET

This implies:

Collisions | Physics Class 11 - NEET

According to the law of conservation of kinetic energy:

Collisions | Physics Class 11 - NEET

This implies:

Collisions | Physics Class 11 - NEET

Dividing equation (iv) by equation (ii):

Collisions | Physics Class 11 - NEET

This implies:

Collisions | Physics Class 11 - NEET

Further, from equation (v) we get:

Collisions | Physics Class 11 - NEET

Substituting this value of v2v_2v2 in equation (i) and rearranging, we get:

Collisions | Physics Class 11 - NEET

Similarly, we get:

Collisions | Physics Class 11 - NEET

Question for Collisions
Try yourself:
In a perfectly inelastic one-dimensional collision between two bodies of equal mass, what happens to their velocities after the collision?
View Solution

Special cases of Head on Elastic Collision

(i) If projectile and target are of the same mass i.e. m_1 = m_2m= m2

Collisions | Physics Class 11 - NEET

(ii) If a massive projectile collides with a light target, i.e. m1 >> m2

Collisions | Physics Class 11 - NEET

Collisions | Physics Class 11 - NEET

(iii) If a light projectile collides with a very heavy target, i.e. m₁ << m₂:

Collisions | Physics Class 11 - NEET

Collisions | Physics Class 11 - NEET

Example 3
Assertion (A): In an elastic collision of two billiard balls, the total kinetic energy is conserved during the short time of collision of the balls (i.e., when they are in contact).

Reason (R): Energy spent against friction does not follow the law of conservation of energy.

(a) Both A and R are true, and R is a correct explanation of A

(b) Both A and R are true, but R is not a correct explanation of A

(c) A is true, but R is false

(d) Both A and R are false

Sol: (d) Both A and R are false

Explanation:

(i) When they are in contact, some part of the kinetic energy may convert into potential energy, so it is not conserved during the short time of collision.

(ii) The law of conservation of energy is always true.

Two-Dimensional or Oblique Collision

If the initial and final velocities of colliding bodies do not lie along the same line, then the collision is called two dimensional or oblique Collision.

In horizontal direction,

m1u1 cos α+ m2u2 cos α2= m1v1 cos β1 + m2v2 cos β2

Collisions | Physics Class 11 - NEET

In vertical direction.

m1u1 sin α1 – m2u2 sin α2 = m1u1 sin β1 – m2u2 sin β2

If m1 = m2 and α1 + α2 = 90°

then β1 + β2 = 90°

If a particle A of mass m1 moving along z-axis with a speed u makes an elastic collision with another stationary body B of mass m2

Collisions | Physics Class 11 - NEET

From conservation law of momentum,

m1u = m1v1 cos α + m2v2 cos β

sin β2v2 sin α – m1v1O = m

Example 4 : Three particles AA, B, and C of equal mass are moving with the same velocity v along the medians of an equilateral triangle. These particles collide at the center G of the triangle. After the collision, A becomes stationary, BB retraces its path with velocity v, then the magnitude and direction of the velocity of CC will be:

(a) vv and opposite to BB
(b) v and in the direction of AA
(c) v and in the direction of C
(d)  v and in the direction of B

Collisions | Physics Class 11 - NEET

Sol: 

Collisions | Physics Class 11 - NEET

Fig 1Fig 1 Fig 2Fig 2

Question for Collisions
Try yourself:
Three particles A, B, and C of equal mass are moving with different velocities along the sides of an equilateral triangle. They collide at the center of the triangle. After the collision, A moves with velocity v in one direction, B moves with velocity v in the opposite direction, what will be the velocity of C?
View Solution

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FAQs on Collisions - Physics Class 11 - NEET

1. What are the main types of collisions in physics?
Ans. The main types of collisions in physics are elastic collisions, inelastic collisions, and perfectly inelastic collisions. In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is conserved, but kinetic energy is not. Perfectly inelastic collisions occur when two objects stick together after colliding, resulting in maximum loss of kinetic energy.
2. How do you calculate the momentum before and after a collision?
Ans. Momentum is calculated using the formula \( p = mv \), where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. To find the total momentum before a collision, sum the momenta of all objects involved. After the collision, use the same formula to calculate the momentum of the combined masses or individual objects, ensuring that momentum conservation principles are applied.
3. What is the difference between elastic and inelastic collisions?
Ans. The primary difference between elastic and inelastic collisions lies in the conservation of kinetic energy. In elastic collisions, both momentum and kinetic energy are conserved, meaning that the total kinetic energy before and after the collision remains the same. In inelastic collisions, momentum is conserved, but kinetic energy is transformed into other forms of energy, such as heat or sound, resulting in a loss of kinetic energy.
4. How does momentum conservation apply to collisions?
Ans. Momentum conservation states that in a closed system with no external forces, the total momentum before a collision is equal to the total momentum after the collision. This principle allows us to analyze collisions by setting the sum of the momenta of the objects before the collision equal to the sum of the momenta after the collision, which is crucial in solving problems related to collisions.
5. What are real-life examples of collisions?
Ans. Real-life examples of collisions include car accidents, sports interactions such as a soccer ball hitting a player's foot, and objects colliding in games like billiards. Each of these scenarios can be analyzed using the principles of physics to understand the forces and momentum involved, helping to improve safety measures or performance.
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