An object of small mass has the same linear momentum as an object of large mass.  Which object has the larger speed?

To determine which object has the larger speed given that they possess the same linear momentum, we can utilize the principle of linear momentum, which is defined mathematically as:

$p = mv$

where:

• $p$ is the momentum,
• $m$ is the mass of the object,
• $v$ is the speed of the object.

1. ### Break Down the Problem

We have two objects:

• Object A (small mass $m_A$)
• Object B (large mass $m_B$)

Both objects have the same momentum $p$: $p_A = p_B$

2. ### Relevant Concepts

For both objects: $p_A = m_A v_A \quad \text{(1)}$ $p_B = m_B v_B \quad \text{(2)}$ Since $p_A = p_B$, we can equate equations (1) and (2): $m_A v_A = m_B v_B$

3. ### Analysis and Detail

Rearranging the equation gives us: $v_A = \frac{m_B}{m_A} v_B \quad \text{(3)}$

Since $m_A < m_B$ (because object A is the small mass and object B is the large mass), the ratio $\frac{m_B}{m_A}$ is greater than 1. This implies that:

$v_A > v_B$

4. ### Verify and Summarize

This means that the object with the smaller mass (object A) must have the larger speed to maintain the same linear momentum as the object with the larger mass (object B).

Final Answer

The object with the smaller mass has the larger speed.