How to Find Momentum After Collision: Elastic, Inelastic, Formula and Problems

The article discusses different formulas and problems on how to find momentum after collision.

An object’s velocity changes during a collision due to external force from another object. The velocity change causes a change in momentum after collision. So, we can find the momentum after collision using the impulse formula, laws of conservation of momentum, and conservation of energy.

The momentum before the collision is Pi =mu. The momentum after collision is also found by estimating a change in an object’s velocity v after the collision. Pf = mv

Read more about Momentum.

Suppose a stationary pull ball having a mass of 8kg is hit by another ball. After the collision, the ball is in motion at 5m/s. Determine the pool ball’s momentum after the collision. 

How to Find Momentum After Collision
How to Find Momentum After Collision

Given:

m = 8kg

v = 5m/s

To Find: ∆P =?

Formula:

∆P = Pf – Pi

Solution:

The momentum of ball after collision is calculated as,

∆P = Pf – Pi

∆P = mv – mu

Since pool ball at rest, i.e., u=0

∆P = mv

Substituting all values,

∆P = 8 x 5

∆P = 40

The pool ball’s momentum after collision is 40kg⋅m/s.

Read more about How to Find Net Force from Momentum.

How to Find Momentum after Collision Formula?

The momentum after collision is determined using the impulse formula.

When we speak about finding momentum after collision of only one object, we can calculate it using the impulse formula. Impulse is the momentum change after collision due to the external force. Since collisions occur rapidly, it is tough to calculate the external force applied and time separately. 

Once we computed momentum before Pi and after collision Pf, we can find impulse in terms of external force by another object as,  

 “Impulse (P) is the product of external force F and time difference (∆t) in which change in momentum occurs.”

What is Impulse
Impulse – Change in Momentum

Mathematically,

∆P = F ∆t

Pf – Pi = F ∆t

Read more about Types of Forces.

A football kicked the football having a mass of 5kg on the frictionless ground surface with a force of 30N over 5 sec. What is the velocity and momentum of football after kicking? 

Given:

m = 5kg

F = 30N

∆t = 5 sec

To Find:

  1. v2=?
  2. Pf=?

Formula:

  1. P = mv
  2. ∆P = F ∆t

Solution:

The momentum of football before kicking is,

Pi = m1v1

Since football is at rest. i.e., v1=0

Therefore, Pi = 0

The momentum of football before kicking is zero.

The momentum of football after kicking is calculated using the Impulse formula.

∆P = F ∆t

Pf-Pi = F ∆t

Since Pi = 0

Pf = F ∆t

Substituting all values,

Pf = 30 x 5

Pf = 150

The momentum of football after kicking is 150kgm/s

The velocity of football after kicking is,

m2v2 = 150

v2 = 150/5

v2 = 30

The velocity of football after kicking is 30m/s.

Read more about How to Find Net Force?

How to Find Total Momentum of Two Objects after Collision?

The total momentum of two objects after collision is estimated using the law of conservation of momentum.

When two objects collide, their respective momentum changes because of their velocities, but their total momentum after collision remains the same. The total momentum after collision is summed by adding all the respective momentums of colliding objects.

In a closed or isolated system, when two objects holding different masses and velocities collide, they may move with each other or away, depending on the types of a collision – such as inelastic collision or elastic collision.

Types of Collision
Elastic and Inelastic Collision
(credit: shutterstock)

After the collision, their momentum, which is the product of their masses and velocities, is also varied. But when talking about the total momentum of an isolated system, it remains unchanged. During the collision, whatever momentum one object loses is gained by another object. That’s how the total momentum of colliding objects is conserved.

Suppose momentum of object 1 is P1 = m1u1

Momentum of object 2 is P2 = m2u2

Momentum of both objects before collision is Pi = P1 + P2 = m1u1 + m2+u2

If there is no net force involved during the collision, then momentum after collision Pf of both objects remains the same as before the collision.  

Therefore, As per law of conservation of momentum,

Pi = Pf

m1u1 + m2+u2 = m1v1 + m2+v2 ……………………. (*)

Notice velocities of both objects changed after collision from u to v. That shows their respective momentum after collision also gets changed.

For an isolated system,

“The total momentum after collision is exactly as before collision as per the law of conservation of momentum.” 

Conservation of Momentum
Total Momentum after Collision
(credit: shutterstock)

Suppose two marble pebbles having masses 10kg and 5kg moving at 8m/sec and 12 m/sec respectively; collide with each other. After the collision, both pebbles move away from each other with the same masses. If one pebble moves away with a velocity of 10m/sec, what is the second pebble’s velocity? 

Given:

m1 = 10kg

m2 = 5kg

u1= 8m/sec

u2= 12m/sec

v1= 10m/sec

To Find: v2 =?

Formula:

m1u1 + m2+u2 = m1v1 + m2+v2

Solution:

The law of conservation of momentum calculates the velocity of the second pebble,

For isolated systems when no net force acts,

m1u1 + m2+u2 = m1v1 + m2+v2

Note that second objects move opposite to the first object. Therefore, the momentum of the second object must be negative. 

Substituting all values,

10 x 8 + (- (5 x12) = 10 x 10 + (-(5xv2)

80 – 60 = 100 -5v2

5v2 = 100 -20

v2 = 80/5

v2 = 16

The velocity of the second pebble after the collision is 16m/sec. 

Read more about Relative Velocity.

How to Find Momentum after Elastic Collision?

The momentum after elastic collision is estimated using the law of conservation of energy. 

The total momentum is conserved during the collision. The kinetic energy of a respective object may change after the collision, but the total kinetic energy after elastic collision stays the same. So, we can find momentum after elastic collision utilizing the law of conservation of energy.

Elastic Collision
How to Find Momentum after Elastic Collision?
(credit: shutterstock)

When the collision between objects is elastic, the total kinetic energy is conserved.

As per law of conservation of energy,

Rearranging equation (*) by terms with m1 on one side and terms with m2 on other. 

Now rearranging equation (#) by terms with m1 on one side and the terms with m2 on other and cancel ½ common factor,

Recognize the first term on the left hand side is ‘1’ in the above equation, we get. 

………………. (1) 

Substitute above equation into equation (*), to eliminate v2, we get

Finally rearrange above equation and solve for velocity v1 of object 1 after collision,

Substitute above equation into equation (1) velocity v2 of object 2 after collision,

Read more about Kinetic Energy.

When a 10kg ball moving at 2m/s elastically collides with another ball having mass 2kg oppositely moving at 4m/s. Calculate the final velocities of both balls after the elastic collision.

Given:

m1 = 10kg

m2 = 2kg

u1 = 2m/s

u2 = -4m/s

To Find:

  1. v1 =?
  2. v2 =?

Formula:

Solution:

The velocity of ball 1 after elastic collision is calculated as,

Substituting all values,

v1 = 0

That means, the elastic collision stopped the ball 1.

The velocity of ball 2 after elastic collision is calculated as,

Substituting all values,

v2= 6 m/s

That means the elastic collision changes the velocity of the second ball to 6m/s.

How to Find Momentum after Inelastic Collision?

The momentum after collision is determined using the law of conservation of momentum.

The total momentum is conserved during the collision. But the total kinetic energy of the system is also changed like the kinetic energy respective object, and the collision is said to be inelastic. So, we can find momentum after inelastic collision using the law of conservation of momentum. 

Inelastic Collision
How to Find Momentum after Inelastic Collision? (credit: shutterstock)

If the collision is elastic, both objects move away from each other with different velocities v1, v2 in opposite directions. 

But if the collision is inelastic, both objects move with one final velocity V in the same direction. 

Therefore, the momentum Pf after inelastic collision becomes m1V + m2V or V(m1+m2)

So, the equation of conservation of momentum for inelastic collision is, 

m1u1 + m2+u2 = V(m1+m2)

The formula for final velocity after inelastic collision is,

V=(m1u1 + m2+u2)/(m1+m2)

Read more about Speed.

Two boys are playing on the playground slide in the park. The first boy having a mass of 20kg sliding at 10m/s on the slide. Since the first boy becomes slower at certain portions latterly collides with another boy having a mass of 30kg who slides down at 12 m/s. What will be the velocity of both boys who slide down together after collision?

Given:

m1 = 20kg

m2 = 30kg

u1 = 10m/s

u2 = 12m/s

To Find: V =?

Formula:

V=(m1u1 + m2+u2)/(m1+m2)

Solution:

The final velocity of both boys sliding after collision is calculated as,

V=(m1u1 + m2+u2)/(m1+m2)

Substituting all values,

V = 11.2

The final velocity of both boys sliding after an inelastic collision is 11.2m/s.


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