Understanding Momentum and its Relationship with Mass and Velocity

This question asks us to compare the momentum of two bodies, A and B, given information about their masses and velocities. To solve this, we need to understand the concept of momentum and its formula.

Momentum is a vector quantity defined as the product of an object's mass and its velocity. It is a measure of the "quantity of motion" an object has. The formula for momentum (\(p\)) is:

$p = m \times v$

where:

• \(p\) is momentum
• \(m\) is mass
• \(v\) is velocity

Analyzing the Given Information about Bodies A and B

We are given two bodies, A and B, with the following characteristics:

• Both bodies A and B are moving with equal velocities. Let's denote this common velocity as \(v\). So, \(v_A = v_B = v\).
• The mass of body B is double that of body A. Let the mass of A be \(m_A\). Then the mass of B is \(m_B = 2m_A\).

Calculating Momentum for Each Body

Now, let's calculate the momentum for each body using the formula \(p = m \times v\).

For body A:

$p_A = m_A \times v_A$

Substituting the given values, we get:

$p_A = m_A \times v$

For body B:

$p_B = m_B \times v_B$

Substituting the given values (\(m_B = 2m_A\) and \(v_B = v\)), we get:

$p_B = (2m_A) \times v$

$p_B = 2 \times m_A \times v$

Comparing the Momenta of A and B

We found that \(p_A = m_A \times v\) and \(p_B = 2 \times m_A \times v\).

Comparing \(p_B\) with \(p_A\), we can see that:

$p_B = 2 \times p_A$

This equation tells us that the momentum of body B is double the momentum of body A.

Evaluating the Options

Let's examine the given options based on our finding that the momentum of B is double that of A.

1. Momentum of B will be double that of A: This matches our calculation \(p_B = 2p_A\). This statement is correct.
2. Momentum of A will be double that of B: This would mean \(p_A = 2p_B\). This is incorrect, as \(p_B = 2p_A\).
3. Momentum of B will be four times that of A: This would mean \(p_B = 4p_A\). This is incorrect, as \(p_B = 2p_A\).
4. Momenta of both A and B will be equal: This would mean \(p_A = p_B\). This is incorrect, as \(p_B = 2p_A\) and \(m_A\) and \(v\) are generally non-zero.

Therefore, the statement that the momentum of B will be double that of A is the correct one.

Conclusion on Momentum Comparison

Since \(p_B = 2m_A v\) and \(p_A = m_A v\), we can conclude that \(p_B = 2 p_A\). The momentum of body B is indeed double that of body A.

Revision Table: Key Concepts in Momentum

Additional Information: Conservation of Momentum

While this question focuses on the definition of momentum, an important related concept is the Law of Conservation of Momentum. This law states that for a closed system (one where no external forces act), the total momentum of the system remains constant. This means that in collisions or explosions within a closed system, the total momentum before the event is equal to the total momentum after the event. This principle is fundamental in physics and has wide applications.

Understanding how mass and velocity affect momentum is crucial for solving problems involving collisions, impulse, and conservation of momentum. In this problem, the direct relationship between mass and momentum (when velocity is constant) is the key. Since body B has twice the mass of body A and they have the same velocity, body B naturally has twice the momentum.