How To Find Conservation Of Momentum: Detailed Explanations

Conservation of momentum is a fundamental principle in physics that states that the total momentum of an isolated system remains constant unless acted upon by an external force. In other words, the momentum of an object or a system does not change unless an external force is applied. This principle is based on Newton’s third law of motion, which states that for every action, there is an equal and opposite reaction.

Understanding and applying the concept of conservation of momentum is crucial in various fields of science and engineering. Whether you are studying mechanics, collisions, or rocket propulsion, conservation of momentum plays a vital role in analyzing and predicting the behavior of objects in motion.

In this blog post, we will explore the formulas, equations, and calculations involved in finding the conservation of momentum, supported by clear examples and explanations.

Formulas and Equations for Conservation of Momentum

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The Basic Equation for Conservation of Momentum

The basic equation for conservation of momentum is:

p_{\text{initial}} = p_{\text{final}}

where p_{\text{initial}} represents the initial momentum of the system, and p_{\text{final}} represents the final momentum of the system.

Derivation of the Conservation of Momentum Equation

The conservation of momentum equation can be derived from Newton’s second law of motion. According to Newton’s second law, the force acting on an object is equal to the rate of change of its momentum. Mathematically, this can be expressed as:

F = \frac{dp}{dt}

where F is the force, dp is the change in momentum, and dt is the change in time.

In an isolated system where no external forces are present, the net force acting on the system is zero. Therefore, the change in momentum is also zero:

\frac{dp}{dt} = 0

This implies that the momentum of the system remains constant, leading to the conservation of momentum equation:

p_{\text{initial}} = p_{\text{final}}

How to Use the Conservation of Momentum Equation

To use the conservation of momentum equation, follow these steps:

  1. Identify the objects or particles involved in the system.
  2. Determine the initial momentum of each object by multiplying its mass (m) by its initial velocity (v): p_{\text{initial}} = m \cdot v.
  3. Determine the final momentum of each object by multiplying its mass (m) by its final velocity (v): p_{\text{final}} = m \cdot v.
  4. Apply the conservation of momentum equation p_{\text{initial}} = p_{\text{final}} to find the relationship between the initial and final velocities of the objects.

Calculating Conservation of Momentum

How to Determine Initial Velocity in Conservation of Momentum

To determine the initial velocity of an object in the conservation of momentum equation, follow these steps:

  1. Identify the object for which you want to find the initial velocity.
  2. Determine the mass of the object (m).
  3. Determine the final velocity of the object (v_{\text{final}}).
  4. Rearrange the conservation of momentum equation p_{\text{initial}} = p_{\text{final}} to solve for the initial velocity (v_{\text{initial}}): v_{\text{initial}} = \frac{p_{\text{final}}}{m}.

How to Measure Final Velocity Using Conservation of Momentum

To measure the final velocity of an object using the conservation of momentum equation, follow these steps:

  1. Identify the object for which you want to measure the final velocity.
  2. Determine the mass of the object (m).
  3. Determine the initial velocity of the object (v_{\text{initial}}).
  4. Rearrange the conservation of momentum equation p_{\text{initial}} = p_{\text{final}} to solve for the final velocity (v_{\text{final}}): v_{\text{final}} = \frac{p_{\text{initial}}}{m}.

How to Calculate Mass in Conservation of Momentum

To calculate the mass of an object in the conservation of momentum equation, follow these steps:

  1. Identify the object for which you want to calculate the mass.
  2. Determine the initial velocity of the object (v_{\text{initial}}).
  3. Determine the final velocity of the object (v_{\text{final}}).
  4. Rearrange the conservation of momentum equation p_{\text{initial}} = p_{\text{final}} to solve for the mass (m): m = \frac{p_{\text{initial}}}{v_{\text{initial}} - v_{\text{final}}}.

Worked Out Examples

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Example of Calculating Initial Velocity

Let’s consider an example where a car with a mass of 1000 kg collides with a stationary object. After the collision, the car’s final velocity is 10 m/s. We can calculate the initial velocity of the car using the conservation of momentum equation.

Given:
– Mass of the car (m): 1000 kg
– Final velocity of the car (v_{\text{final}}): 10 m/s

Using the conservation of momentum equation p_{\text{initial}} = p_{\text{final}}, we can solve for the initial velocity (v_{\text{initial}}) of the car:

v_{\text{initial}} = \frac{p_{\text{final}}}{m} = \frac{1000 \, \text{kg} \cdot 10 \, \text{m/s}}{1000 \, \text{kg}} = 10 \, \text{m/s}

Therefore, the initial velocity of the car is 10 m/s.

Example of Determining Final Velocity

Consider a neutron with a mass of 1 kg moving with an initial velocity of 5 m/s. It collides with a stationary car with a mass of 2000 kg. We can determine the final velocity of the neutron using the conservation of momentum equation.

Given:
– Mass of the neutron (m): 1 kg
– Initial velocity of the neutron (v_{\text{initial}}): 5 m/s
– Mass of the car (m): 2000 kg

Using the conservation of momentum equation p_{\text{initial}} = p_{\text{final}}, we can solve for the final velocity (v_{\text{final}}) of the neutron:

v_{\text{final}} = \frac{p_{\text{initial}}}{m} = \frac{(1 \, \text{kg} \cdot 5 \, \text{m/s})}{1 \, \text{kg} + 2000 \, \text{kg}} \approx 0.0025 \, \text{m/s}

Therefore, the final velocity of the neutron is approximately 0.0025 m/s.

Example of Measuring Mass Using Conservation of Momentum

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Suppose a car with an initial velocity of 20 m/s collides with a stationary object, resulting in a final velocity of 10 m/s. We can calculate the mass of the object using the conservation of momentum equation.

Given:
– Initial velocity of the car (v_{\text{initial}}): 20 m/s
– Final velocity of the car (v_{\text{final}}): 10 m/s

Using the conservation of momentum equation p_{\text{initial}} = p_{\text{final}}, we can solve for the mass (m) of the object:

m = \frac{p_{\text{initial}}}{v_{\text{initial}} - v_{\text{final}}} = \frac{(m \cdot v_{\text{initial}})}{(v_{\text{initial}} - v_{\text{final}})}

Simplifying the equation, we find:

v_{\text{initial}} - v_{\text{final}} = \frac{m \cdot v_{\text{initial}}}{m} = v_{\text{initial}}

Therefore, the mass of the object is equal to the initial velocity of the car: m = 20 \, \text{kg}.

Understanding and applying the concept of conservation of momentum is crucial in analyzing the behavior of objects in motion. By following the conservation of momentum equation and the associated calculations, we can determine initial velocity, measure final velocity, and calculate mass in various scenarios. Remember to always apply the principle of conservation of momentum to isolated systems and consider the effects of external forces when necessary.

Numerical Problems on how to find conservation of momentum

how to find conservation of momentum
Image by Free High School Science Texts Authors – Wikimedia Commons, Licensed under CC BY-SA 3.0.

Problem 1:

how to find conservation of momentum
Image by No-w-ay – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 4.0.

A car of mass 1000 kg is initially at rest. It collides with a stationary truck of mass 4000 kg. After the collision, the car moves with a velocity of 20 m/s to the right. What is the velocity of the truck after the collision?

Solution:

Let the velocity of the truck after the collision be v (to the right).

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
Total momentum = (mass of car) x (velocity of car) + (mass of truck) x (velocity of truck)
= (1000 kg) x (0 m/s) + (4000 kg) x (0 m/s)
= 0 kg*m/s

After the collision:
Total momentum = (mass of car) x (velocity of car) + (mass of truck) x (velocity of truck)
= (1000 kg) x (20 m/s) + (4000 kg) x (v m/s)
= 20000 kgm/s + 4000v kgm/s

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
0 kgm/s = 20000 kgm/s + 4000v kg*m/s

Solving for v:
4000v kgm/s = -20000 kgm/s
v = -5 m/s

Therefore, the velocity of the truck after the collision is 5 m/s to the left.

Problem 2:

A bullet of mass 0.02 kg is fired horizontally into a wooden block of mass 0.5 kg initially at rest on a smooth surface. After the collision, the bullet gets embedded in the block and the block moves with a velocity of 2 m/s. What was the initial velocity of the bullet?

Solution:

Let the initial velocity of the bullet be u.

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
Total momentum = (mass of bullet) x (velocity of bullet) + (mass of block) x (velocity of block)
= (0.02 kg) x (u m/s) + (0.5 kg) x (0 m/s)
= 0.02u kg*m/s

After the collision:
Total momentum = (mass of bullet) x (velocity of bullet) + (mass of block) x (velocity of block)
= (0.02 kg) x (0 m/s) + (0.5 kg) x (2 m/s)
= 1 kg*m/s

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
0.02u kgm/s = 1 kgm/s

Solving for u:
0.02u = 1
u = 50 m/s

Therefore, the initial velocity of the bullet was 50 m/s.

Problem 3:

Two objects of masses 3 kg and 5 kg, initially at rest, collide and stick together. If the final velocity of the combined object is 2 m/s, what is the velocity of the 3 kg object before the collision?

Solution:

Let the velocity of the 3 kg object before the collision be u.

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
Total momentum = (mass of 3 kg object) x (velocity of 3 kg object) + (mass of 5 kg object) x (velocity of 5 kg object)
= (3 kg) x (u m/s) + (5 kg) x (0 m/s)
= 3u kg*m/s

After the collision:
Total momentum = (mass of combined object) x (velocity of combined object)
= (3 kg + 5 kg) x (2 m/s)
= 16 kg*m/s

According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
3u kgm/s = 16 kgm/s

Solving for u:
3u = 16
u = 16/3 m/s

Therefore, the velocity of the 3 kg object before the collision is 16/3 m/s.

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