Intro to Conservation of Momentum: Videos & Practice Problems

Momentum is conserved in a system of interacting objects, meaning the total momentum before an interaction equals the total momentum after. The conservation of momentum equation is expressed as $p_{\mathrm{system}}=p_1+p_2$. This principle applies to collisions, where the direction of velocities must be considered, ensuring accurate calculations of final velocities and directions post-collision.

Total Momentum of a System of Objects

Understanding momentum is crucial when analyzing interactions between objects, particularly during collisions. The principle of momentum conservation states that in a closed system, the total momentum remains constant when two or more objects interact. A system is simply a collection of objects, which can range from one to several, depending on the problem at hand.

To calculate the total momentum of a system comprising multiple objects, we focus on the individual momenta of each object. The momentum (p) of an object is defined as the product of its mass (m) and velocity (v), expressed mathematically as:

\[p = mv\]

For a system with two objects, the total momentum can be represented as:

\[p_{\text{system}} = p_1 + p_2\]

Substituting the momentum formula, this becomes:

\[p_{\text{system}} = m_1 v_1 + m_2 v_2\]

In practice, when calculating the total momentum, it is essential to consider the direction of each object's velocity. For instance, if one object moves to the right and another to the left, we must assign a positive direction to one of them. In this case, if we designate right as positive, the leftward velocity will be negative.

For example, consider two objects: object A with a mass of 4 kg moving to the right at 12 m/s, and object B with a mass of 5 kg moving to the left at 9 m/s. The momentum of each object can be calculated as follows:

For object A:

\[p_A = m_A v_A = 4 \, \text{kg} \times 12 \, \text{m/s} = 48 \, \text{kg m/s}\]

For object B (noting the negative velocity):

\[p_B = m_B v_B = 5 \, \text{kg} \times (-9 \, \text{m/s}) = -45 \, \text{kg m/s}\]

Now, summing these momenta gives:

\[p_{\text{system}} = 48 \, \text{kg m/s} + (-45 \, \text{kg m/s}) = 3 \, \text{kg m/s}\]

This result indicates that the total momentum of the system is 3 kg m/s to the right, demonstrating how the individual momenta combine to yield a net momentum for the system. Understanding these calculations and the importance of direction is vital for solving momentum-related problems effectively.

Object A moves at 10 m/s at 53° and Object B moves at 5 m/s at –37° as shown below. Calculate the magnitude of the system's total momentum if both objects have a mass of 2kg.

• A. $21.6\operatorname{kg}\cdot\frac{m}{s}$
• B. $22.4\operatorname{kg}\cdot\frac{m}{s}$
• C. $29.7\operatorname{kg}\cdot\frac{m}{s}$
• D. $9.17\operatorname{kg}\cdot\frac{m}{s}$

Answer: (B)

Conservation Of Momentum

In physics, understanding the concept of momentum is crucial, especially when dealing with systems of interacting objects. Momentum, denoted as p, is defined as the product of an object's mass and its velocity, expressed mathematically as p = mv. When multiple objects are involved, the total momentum of the system, referred to as p_system, is the sum of the individual momenta of all objects in the system.

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act upon it. This can be expressed with the equation:

p_{system initial} = p_{system final}

For two objects, this can be rewritten as:

p_{1 initial} + p_{2 initial} = p_{1 final} + p_{2 final}

Substituting the momentum formula, we have:

m₁v_{1 initial} + m₂v_{2 initial} = m₁v_{1 final} + m₂v_{2 final}

This equation is fundamental in solving problems involving collisions or interactions between objects. To illustrate this, consider a scenario with two balls colliding. For example, let’s say we have a 3 kg ball moving to the right at 7 m/s and a 4 kg ball moving to the left at 5 m/s. To analyze the situation, we first assign a positive direction (to the right) and a negative direction (to the left). Thus, the initial velocities can be represented as:

v_{A initial} = 7 m/s (positive)

v_{B initial} = -5 m/s (negative)

After the collision, if the 4 kg ball moves to the right at 2 m/s, we can denote its final velocity as:

v_{B final} = 2 m/s (positive)

To find the final velocity of the 3 kg ball, we set up the conservation of momentum equation:

3 kg * 7 m/s + 4 kg * (-5 m/s) = 3 kg * v_{A final} + 4 kg * 2 m/s

Calculating the left side gives:

21 kg·m/s - 20 kg·m/s = 1 kg·m/s

On the right side, we have:

3 kg * v_{A final} + 8 kg·m/s

Setting the two sides equal:

1 kg·m/s = 3 kg * v_{A final} + 8 kg·m/s

Rearranging to solve for v_{A final} gives:

1 - 8 = 3 v_{A final}

-7 = 3 v_{A final}

v_{A final} = -\frac{7}{3} m/s ≈ -2.33 m/s

The negative sign indicates that the final velocity of the 3 kg ball is directed to the left. Thus, the final result shows that after the collision, the 3 kg ball moves to the left at approximately 2.33 m/s. This example illustrates how the conservation of momentum can be applied to determine the outcomes of collisions in a straightforward manner.

On a frictionless air hockey table, puck A of mass 0.250 kg moves to the right and collides with puck B of mass 0.38kg, which is initially at rest. After the collision, puck A is moving the left at 0.12 m/s and puck B moves to the right at 0.65 m/s. What was the initial velocity of puck A before the collision?

• A. 0.73 m/s
• B. 0.57 m/s
• C. 0.87 m/s
• D. 1.1 m/s

Answer: (C)

Here’s what students ask on this topic:

What is the principle of conservation of momentum?

The principle of conservation of momentum states that in a closed system with no external forces, the total momentum remains constant before and after an interaction. Mathematically, this is expressed as:

$p_{\mathrm{system}}=p_1+p_2$

This means that the sum of the momenta of all objects in the system before the interaction is equal to the sum of the momenta after the interaction. This principle is crucial in analyzing collisions and interactions in physics.

How do you calculate the total momentum of a system of objects?

To calculate the total momentum of a system of objects, you sum the individual momenta of each object. The momentum of an object is given by the product of its mass and velocity:

$p=mv$

For a system with two objects, the total momentum is:

$p_{\mathrm{system}}=m_1{v}_1+m_2{v}_2$

Remember to consider the direction of velocities, assigning positive or negative signs accordingly.

What does it mean for momentum to be conserved in a collision?

When momentum is conserved in a collision, it means that the total momentum of the system before the collision is equal to the total momentum after the collision. This is expressed as:

$p_{\mathrm{initial}}=p_{\mathrm{final}}$

For two objects, this can be written as:

$m_1{v}_1{\mathrm{initial}}_{}+m_2{v}_2{\mathrm{initial}}_{}=m_1{v}_1{\mathrm{final}}_{}+m_2{v}_2{\mathrm{final}}_{}$

This principle helps in determining the final velocities of objects after a collision.

How do you solve a problem using the conservation of momentum equation?

To solve a problem using the conservation of momentum equation, follow these steps:

1. Draw a diagram of the system before and after the interaction.
2. Write the conservation of momentum equation:

$m_1{v}_1{\mathrm{initial}}_{}+m_2{v}_2{\mathrm{initial}}_{}=m_1{v}_1{\mathrm{final}}_{}+m_2{v}_2{\mathrm{final}}_{}$

3. Substitute the known values for masses and velocities.
4. Solve for the unknown variable, keeping track of the signs for direction.

This method ensures that the total momentum before and after the interaction is equal, allowing you to find the unknown quantities.

Why is it important to consider the direction of velocities in momentum problems?

Considering the direction of velocities in momentum problems is crucial because momentum is a vector quantity, meaning it has both magnitude and direction. When calculating the total momentum of a system, you must account for the direction to ensure accurate results. Assigning positive and negative signs to velocities based on their direction helps in correctly summing the momenta. For example, if one object moves to the right and another to the left, their momenta will have opposite signs, affecting the total momentum calculation. Ignoring direction can lead to incorrect conclusions about the system's behavior.