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 P_{i} =mu. The momentum after collision is also found by estimating a change in an object’s velocity v after the collision. P_{f} = mv

**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. **

** Given**:

m = 8kg

v = 5m/s

** To Find**: ∆P =?

** Formula**:

∆P = P_{f} – P_{i}

** Solution**:

The momentum of ball after collision is calculated as,

∆P = P_{f} – P_{i}

∆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 P_{i} and after collision P_{f}, 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.”*

Mathematically,

∆P = F ∆t

P_{f} – P_{i} = 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**:

- v
_{2}=? - P
_{f}=?

** Formula**:

- P = mv
- ∆P = F ∆t

** Solution**:

The momentum of football before kicking is,

P_{i} = m_{1}v_{1}

Since football is at rest. i.e., v_{1}=0

Therefore, P_{i} = 0

The momentum of football before kicking is zero.

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

∆P = F ∆t

P_{f}-P_{i} = F ∆t

Since Pi = 0

P_{f} = F ∆t

Substituting all values,

P_{f} = 30 x 5

P_{f} = 150

**The momentum of football after kicking is 150kg****⋅****m/s**

The velocity of football after kicking is,

m_{2}v_{2} = 150

v_{2} = 150/5

v_{2} = 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.**

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 P_{1} = m_{1}u_{1}

Momentum of object 2 is P_{2} = m_{2}u_{2}

Momentum of both objects before collision is P_{i} = P_{1} + P_{2} = m_{1}u_{1} + m_{2}+u_{2}

**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**,

P_{i} = P_{f}

m_{1}u_{1} + m_{2}+u_{2} = m_{1}v_{1} + m_{2}+v_{2} ……………………. (*)

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.” **

**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**:

m_{1} = 10kg

m_{2} = 5kg

u_{1}= 8m/sec

u_{2}= 12m/sec

v_{1}= 10m/sec

** To Find**: v

_{2}=?

** Formula**:

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

** Solution**:

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

For isolated systems when no net force acts,

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

__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

5v_{2} = 100 -20

v_{2} = 80/5

v_{2} = 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.**

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 v_{2}, we get

Finally rearrange above equation and solve for **velocity v _{1} of object 1 after collision**,

Substitute above equation into equation (1) **velocity v _{2} 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**:

m_{1} = 10kg

m_{2} = 2kg

u_{1} = 2m/s

u_{2} = -4m/s

** To Find**:

- v
_{1}=? - v
_{2}=?

** Formula**:

** Solution**:

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

Substituting all values,

v_{1} = 0

**That means, the elastic collision stopped the ball 1.**

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

Substituting all values,

v_{2}= 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. **

If the collision is elastic, both objects move away from each other with different velocities v_{1}, v_{2} in opposite directions.

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

Therefore, the momentum P_{f} after inelastic collision becomes m_{1}V + m_{2}V or V(m_{1}+m_{2})

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

m_{1}u_{1} + m_{2}+u_{2 }= V(m_{1}+m_{2})

The formula for **final**** velocity after inelastic collision** is,

V=(m_{1}u_{1} + m_{2}+u_{2})/(m_{1}+m_{2})

**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**:

m_{1} = 20kg

m_{2} = 30kg

u_{1} = 10m/s

u_{2} = 12m/s

** To Find**: V =?

** Formula**:

V=(m_{1}u_{1} + m_{2}+u_{2})/(m_{1}+m_{2})

** Solution**:

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

V=(m_{1}u_{1} + m_{2}+u_{2})/(m_{1}+m_{2})

Substituting all values,

V = 11.2

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