Adding Mass to a Moving System: Videos & Practice Problems

In a system where mass is added to a moving object, such as a sled with a box dropped onto it, momentum conservation is key. The final velocity of the combined system can be calculated using the equation m_1 × ⁢v_1i + m_2 × ⁢v_2i = (m+m)⁢vf. The sled's speed decreases as mass increases, demonstrating the principle of momentum transfer, where one object's gain in momentum equals another's loss.

Adding Mass to a Moving System

In problems involving the addition of mass to a moving system, such as a sled with a box dropped onto it, the principles of momentum conservation play a crucial role. When two objects combine, they must move together with a common final velocity. This scenario is akin to a completely inelastic collision, where the objects stick together after the interaction.

Consider a sled with a mass of 70 kg moving to the right at a speed of 10 m/s. When a 30 kg box is dropped onto the sled, we need to determine the final speed of the combined system. The momentum conservation equation can be expressed as:

$$m_1 v_{1\text{ initial}} + m_2 v_{2\text{ initial}} = (m_1 + m_2) v_{\text{final}}$$

Here, \(m_1\) is the mass of the sled, \(v_{1\text{ initial}}\) is its initial velocity, \(m_2\) is the mass of the box, and \(v_{2\text{ initial}}\) is the initial velocity of the box. Since the box is dropped vertically, it has no initial horizontal velocity, allowing us to simplify the equation to:

$$70 \times 10 + 30 \times 0 = (70 + 30) v_{\text{final}}$$

Calculating this gives:

$$700 = 100 v_{\text{final}}$$

Thus, the final velocity \(v_{\text{final}}\) is:

$$v_{\text{final}} = 7 \text{ m/s}$$

This result indicates that the sled's speed decreases from 10 m/s to 7 m/s upon the addition of the box, demonstrating that as mass increases, speed must decrease to conserve momentum.

Next, we can calculate the change in momentum for both the box and the sled. The change in momentum (\(\Delta p\)) for the box is given by:

$$\Delta p_2 = m_2 v_{\text{final}} - m_2 v_{\text{initial}}$$

Substituting the values, we find:

$$\Delta p_2 = 30 \times 7 - 30 \times 0 = 210 \text{ kg m/s}$$

For the sled, the change in momentum is calculated similarly:

$$\Delta p_1 = m_1 v_{\text{final}} - m_1 v_{\text{initial}}$$

Substituting the sled's values gives:

$$\Delta p_1 = 70 \times 7 - 70 \times 10 = -210 \text{ kg m/s}$$

These results show that the box gains momentum while the sled loses an equal amount, illustrating the principle of momentum conservation: if one object gains momentum, another must lose the same amount. This exchange is fundamental in understanding how systems interact and respond to changes in mass and velocity.

A 40-kg skater runs parallel to a 3-kg skateboard. Both are moving to the right at 10m/s. The skater jumps on the board, and they move continue moving right (i.e. no change in direction). Calculate the final speed of the system.

• A. 0.7 m/s
• B. 3 m/s
• C. 6 m/s
• D. 10 m/s

Answer: (D)

Here’s what students ask on this topic:

What is the final velocity of a system when mass is added to a moving object?

The final velocity of a system when mass is added to a moving object can be calculated using the principle of momentum conservation. The equation is:

$m_1{v}_1+m_2{v}_2=(m_1+m_2)v_f$

Here, $m_1$ and $m_2$ are the masses of the objects, and $v_1$ and $v_2$ are their initial velocities. $v_f$ is the final velocity of the combined system. By solving for $v_f$, you can determine the final velocity after the mass is added.

How does adding mass to a moving system affect its velocity?

Adding mass to a moving system decreases its velocity. This is due to the principle of momentum conservation, which states that the total momentum of a system remains constant if no external forces act on it. The equation for momentum conservation is:

$m_1{v}_1+m_2{v}_2=(m_1+m_2)v_f$

When mass is added, the total mass increases, and to keep the momentum constant, the velocity must decrease proportionally.

What is the principle of momentum conservation in the context of adding mass to a moving system?

The principle of momentum conservation states that the total momentum of a closed system remains constant if no external forces act on it. In the context of adding mass to a moving system, this principle implies that the initial momentum of the system (before adding mass) must equal the final momentum (after adding mass). The equation is:

$m_1{v}_1+m_2{v}_2=(m_1+m_2)v_f$

Here, $m_1$ and $m_2$ are the masses, and $v_1$ and $v_2$ are their initial velocities. $v_f$ is the final velocity of the combined system.

How do you calculate the change in momentum when mass is added to a moving system?

The change in momentum when mass is added to a moving system can be calculated using the formula:

$\Delta p=m_2{v}_f-m_2{v}_2$

Here, $m_2$ is the mass added, $v_f$ is the final velocity of the system, and $v_2$ is the initial velocity of the added mass. The change in momentum is the difference between the final and initial momentum of the added mass.

Why does the velocity decrease when mass is added to a moving system?

The velocity decreases when mass is added to a moving system due to the principle of momentum conservation. The total momentum of the system must remain constant if no external forces act on it. The equation for momentum conservation is:

$m_1{v}_1+m_2{v}_2=(m_1+m_2)v_f$

When mass is added, the total mass increases. To keep the momentum constant, the velocity must decrease proportionally. This ensures that the product of mass and velocity (momentum) remains unchanged.