Problem 51 - College Physics

Question

\(\cdot\) A 1200 kg station wagon is moving along a straight high-way at 12.0 \(\mathrm{m} / \mathrm{s}\) . Another car, with mass 1800 \(\mathrm{kg}\) and speed \(20.0 \mathrm{m} / \mathrm{s},\) has its center of mass 40.0 \(\mathrm{m}\) ahead of the center of mass of the station wagon. (See Figure 8.43.) (a) Find the position of the center of mass of the system consisting of the two automobiles. (b) Find the magnitude of the total momentum of the system from the given data. (c) Find the speed of the center of mass of the system. (d) Find the total momentum of the system, using the speed of the center of mass. Compare your result with that of part (b)

Step-by-Step Solution

Answer

(a) 24 m from station wagon center, (b) 48,000 kg m/s, (c) 16 m/s, (d) 48,000 kg m/s, consistent with (b).

Step 1: Understand the Problem

We need to find the center of mass of the system consisting of two cars, and calculate various properties like the total momentum and speed of the center of mass by using their respective masses, speeds, and positions. The data includes two cars' mass and speed, and their initial separation of centers.

Step 2: Calculate the Center of Mass Position

The position of the center of mass for two objects is given by \( \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \), where \(m_1\) and \(m_2\) are masses of the station wagon and the second car, and \(x_1\) and \(x_2\) are their positions. Assuming the station wagon to be at the origin, \(x_1 = 0 \) and \( x_2 = 40.0 \, \text{m} \). Therefore, the center of mass position is \( \frac{1200 \times 0 + 1800 \times 40.0}{1200 + 1800} \).

Step 3: Calculate the Total Momentum of the System

The total momentum of the system is given by the sum of the momentum of each car, \( p = m_1v_1 + m_2v_2 \). So, for the cars, it is \( 1200 \times 12.0 + 1800 \times 20.0 \).

Step 4: Calculate the Speed of the Center of Mass

The speed of the center of mass is given by \( \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \). Substitute the known values to get \( \frac{1200 \times 12.0 + 1800 \times 20.0}{1200 + 1800} \).

Step 5: Calculate Total Momentum Using the Center of Mass Speed

The total momentum should also equal \( v_{\text{cm}}(m_1 + m_2) \), where \( v_{\text{cm}} \) is the speed of the center of mass found in the previous step. Verify this matches the result from Step 3.

Key Concepts

Momentum

Momentum is a fundamental concept in physics that refers to the quantity of motion an object possesses. It is determined by the product of an object's mass and velocity. The formula for momentum is simple yet powerful:

• \( p = m imes v \)

where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.
In our example with two cars, we consider each car's mass and velocity to calculate their individual momenta. For the station wagon, this would be \(1200 \times 12.0\). For the second car, it is \(1800 \times 20.0\).
By adding these individual momenta, we get the total momentum of the system:

• \( p_{ ext{total}} = p_1 + p_2 \)

This cumulative momentum accounts for the motion characteristics of both vehicles as a combined system.

Speed Calculation

When two objects are considered as a system, finding their combined speed of motion can help in understanding how the system behaves as a whole. For the speed of the center of mass, we utilize the weighted average of the speeds of the constituent objects based on their masses.
The formula used is:

• \( v_{ ext{cm}} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \)

where \(v_{cm}\) is the speed of the center of mass, \(m_1, v_1\) are the mass and speed for the first object, and \(m_2, v_2\) are for the second.
In this exercise, the speed of the center of mass is crucial for understanding how two cars move together as a system, which changes slowly due to their combined weight and distribution of velocity.

Mass Distribution

The distribution of mass within a system impacts its center of mass, which represents the average location of the mass. This is particularly useful in determining how systems move and balance.
When calculating the center of mass position for two cars in a straight line, we assume a reference point, often choosing one object as the origin. The formula:

• \( x_{ ext{cm}} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \)

considers not only the individual positions \((x_1, x_2)\) but also their weights \((m_1, m_2)\).
This mass distribution formula is valuable for knowing how the combined system will react when forces are applied, revealing the dynamics of the objects' interactions.

Physics Problem Solving

Tackling physics problems effectively requires clear understanding and strategic application of underlying concepts such as momentum, speed calculation, and mass distribution. It is often best to follow a methodical step-by-step approach:

• Understand: Identify what you need to find or prove.
• List knowns: Gather given data like mass, velocity, and distances.
• Use equations: Apply relevant formulas to connect the data to the unknowns.
• Verify: Ensure results match expected physical behavior or constraints.

This process allows one to solve even complex physics problems efficiently. In problems like determining the center of mass and momentum, clear application of physics principles simplifies what might seem daunting initially.