# How to Find Momentum with Conservation Laws: A Comprehensive Guide

## Summary

The principle of conservation of momentum states that the total momentum of a closed system remains constant before and after a collision or interaction, provided no external forces are acting on the system. This principle can be used to calculate the final velocities of objects involved in a collision by applying the mathematical equation: m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f, where m1, m2 are the masses of the objects, and v1i, v2i, v1f, v2f are the initial and final velocities, respectively. This guide will provide a detailed explanation of the conservation of momentum principle, along with examples, physics formulas, numerical problems, and experimental verification using a linear air track.

## Understanding the Principle of Conservation of Momentum

The principle of conservation of momentum is a fundamental law in physics that states that the total momentum of a closed system remains constant before and after a collision or interaction, provided no external forces are acting on the system. This means that the total momentum of the system is conserved, and the sum of the initial momenta is equal to the sum of the final momenta.

Mathematically, the principle of conservation of momentum can be represented as:

m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

Where:
– m1 and m2 are the masses of the objects
– v1i and v2i are the initial velocities of the objects
– v1f and v2f are the final velocities of the objects

This equation can be used to calculate the final velocities of the objects involved in a collision if their initial velocities and masses are known.

## Applying the Conservation of Momentum Principle

To apply the conservation of momentum principle, let’s consider an example:

Suppose two objects with masses m1 = 2 kg and m2 = 3 kg have initial velocities v1i = 5 m/s and v2i = 0 m/s, respectively. If the two objects stick together after the collision, we can find the final velocity (v) of the combined object using the conservation of momentum principle:

m1 * v1i + m2 * v2i = (m1 + m2) * v
2 * 5 + 3 * 0 = (2 + 3) * v
10 = 5 * v
v = 2 m/s

Therefore, the final velocity of the combined object is 2 m/s.

## Physics Formulas and Equations

The conservation of momentum principle can be expressed using the following formula:

p = m * v

Where:
– p is the momentum of the object
– m is the mass of the object
– v is the velocity of the object

The total momentum of a system is the sum of the individual momenta of the objects in the system:

P_total = p1 + p2 + … + pn
P_total = m1 * v1 + m2 * v2 + … + mn * vn

The conservation of momentum principle can be written as:

P_initial = P_final
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

## Numerical Problems

1. Two objects with masses m1 = 3 kg and m2 = 5 kg have initial velocities v1i = 4 m/s and v2i = -2 m/s, respectively. After a collision, the final velocities of the objects are v1f = 2 m/s and v2f = 0 m/s. Calculate the total momentum before and after the collision.

Given:
– m1 = 3 kg, m2 = 5 kg
– v1i = 4 m/s, v2i = -2 m/s
– v1f = 2 m/s, v2f = 0 m/s

Initial total momentum:
P_initial = m1 * v1i + m2 * v2i
P_initial = 3 * 4 + 5 * (-2)
P_initial = 12 – 10
P_initial = 2 kg·m/s

Final total momentum:
P_final = m1 * v1f + m2 * v2f
P_final = 3 * 2 + 5 * 0
P_final = 6 kg·m/s

The total momentum is conserved, as P_initial = P_final = 2 kg·m/s.

1. A 2 kg object moving at 5 m/s collides with a 3 kg object moving at 3 m/s in the opposite direction. If the two objects stick together after the collision, find the final velocity of the combined object.

Given:
– m1 = 2 kg, m2 = 3 kg
– v1i = 5 m/s, v2i = -3 m/s

Using the conservation of momentum principle:
m1 * v1i + m2 * v2i = (m1 + m2) * v
2 * 5 + 3 * (-3) = (2 + 3) * v
10 – 9 = 5 * v
1 = 5 * v
v = 0.2 m/s

Therefore, the final velocity of the combined object is 0.2 m/s.

## Experimental Verification using a Linear Air Track

To experimentally verify the conservation of momentum principle, we can use a linear air track, which provides linear and stable motion with negligible friction force between the vehicles and the track. By colliding two vehicles on the track and using photographic techniques to obtain the instantaneous velocities of the objects before and after the collision, we can verify the conservation of momentum law.

The experimental setup typically includes the following components:
– Linear air track
– Two vehicles with known masses
– High-speed camera or motion sensor to measure the velocities
– Collision detection system

The general procedure for the experiment is as follows:
1. Measure the masses of the two vehicles.
2. Set the initial velocities of the vehicles on the linear air track.
3. Initiate the collision and use the high-speed camera or motion sensor to record the instantaneous velocities of the vehicles before and after the collision.
4. Calculate the total momentum before and after the collision using the measured velocities and masses.
5. Verify that the total momentum is conserved, within the experimental uncertainties.

The experiment shows that the percentages of accuracy are up to 90%, indicating that the linear air track can be used to verify the linear momentum conservation law on any collision between two objects.

## Conclusion

In summary, the principle of conservation of momentum is a fundamental law in physics that states that the total momentum of a closed system remains constant before and after a collision or interaction, provided no external forces are acting on the system. This principle can be used to calculate the final velocities of objects involved in a collision by applying the mathematical equation: m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f.

The conservation of momentum principle can be expressed using the formula p = m * v, and the total momentum of a system is the sum of the individual momenta of the objects in the system. Numerical problems and experimental verification using a linear air track can be used to demonstrate the conservation of momentum principle.