Problem 46 - College Physics

Question

A \(1200 \mathrm{~kg}\) station wagon is moving along a straight highway 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 \(8.46 .\) ) (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) The center of mass is at 24 m. (b) Total momentum is 50400 kg m/s. (c) Speed of center of mass is 16.8 m/s. (d) Total momentum, using speed of center of mass, is 50400 kg m/s, consistent with (b).

Step 1: Understand the Problem

We need to find the center of mass, total momentum, and speed of the center of mass for a system of two moving cars with different masses and velocities.

Step 2: Center of Mass Calculation

The center of mass \( x_{cm} \) of two objects can be found by the formula:\[x_{cm} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2}\]where \( m_1 = 1200 \text{ kg}, x_1 = 0 \text{ m} \) (position of station wagon), \( m_2 = 1800 \text{ kg}, x_2 = 40 \text{ m} \) (position of the second car). Substitute these values into the formula:\[x_{cm} = \frac{1200 \cdot 0 + 1800 \cdot 40}{1200 + 1800} = \frac{72000}{3000} = 24 \text{ m}\]

Step 3: Total Momentum of the System

The total momentum \( P_{total} \) is the sum of the momentum of each car:\[P_{total} = m_1 \cdot v_1 + m_2 \cdot v_2\]where \( m_1 = 1200 \text{ kg}, v_1 = 12 \text{ m/s} \), and \( m_2 = 1800 \text{ kg}, v_2 = 20 \text{ m/s} \). Substitute these values:\[P_{total} = 1200 \cdot 12 + 1800 \cdot 20 = 14400 + 36000 = 50400 \text{ kg m/s}\]

Step 4: Speed of the Center of Mass

The speed of the center of mass \( v_{cm} \) can be calculated by the formula:\[v_{cm} = \frac{m_1 \cdot v_1 + m_2 \cdot v_2}{m_1 + m_2}\]Using the given values:\[v_{cm} = \frac{1200 \cdot 12 + 1800 \cdot 20}{1200 + 1800} = \frac{50400}{3000} = 16.8 \text{ m/s}\]

Step 5: Compare Momentum Calculations

We re-calculate total momentum using the speed of the center of mass:\[P'_{total} = (m_1 + m_2) \cdot v_{cm} = 3000 \cdot 16.8 = 50400 \text{ kg m/s}\]This is consistent with our part (b) calculation, proving our calculations are correct.

Key Concepts

Center of Mass

When we talk about the center of mass, we're discussing the point where the total mass of a system seems to be concentrated. For the two cars in this problem, the station wagon and the second car, their individual masses and positions allow us to find a shared balance point. Imagine holding a long bar with weights at each end: the center of mass is where you can balance it on one finger.
The formula for the center of mass is given by the equation:

• \(x_{cm} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2}\)

In this equation, \(m_1\) and \(m_2\) are the masses of the station wagon and the other car, while \(x_1\) and \(x_2\) are their respective positions. We've found this center to be 24 meters from the position of the station wagon. These numbers mean that the cars balance out at this point along the road.

Momentum

Momentum is a measure of the quantity of motion an object has. If you've ever seen a bowling ball hurtle down a lane, the ball has momentum — a product of its mass and velocity. For our cars, the total momentum of the system is found by adding the momentum of each car. Momentum (\(P\)) is calculated as:

• \(P = m \cdot v\)

where \(m\) is mass, and \(v\) is velocity. For this problem, the station wagon and the second car have masses of 1200 kg and 1800 kg, and velocities of 12 m/s and 20 m/s, respectively.
Their combined momentum is

• \(P_{total} = 50400 \text{ kg m/s}\).

This tells us how much force, roughly speaking, this car system could exert if it stopped suddenly.

Velocity

Velocity is a vector quantity that indicates speed with a direction. In simpler terms, it's how fast something's going and where it's headed. For our car system, we're interested in the velocity of the center of mass.The speed of the center of mass \(v_{cm}\) can be thought of as the average speed when taking into account the masses and velocities of both cars. Using the formula:

• \(v_{cm} = \frac{m_1 \cdot v_1 + m_2 \cdot v_2}{m_1 + m_2}\)

we find the system's center of mass to be moving at 16.8 m/s. This speed represents a "weighted average" of the two cars' speeds, where each car's influence is based on its respective mass.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It's one of the fundamental concepts in mechanics. For each of our cars, kinetic energy is measured by:

• \(KE = \frac{1}{2} m v^2\)

This expresses how much work the car can perform due to its speed. In a moving car, this energy is what enables it to continue moving unless an opposing force, like braking or friction, acts on it. It's fascinating to note that even if momentum and velocity might be combined, kinetic energy emphasizes the object's speed more because of the squared velocity in its formula.
Computing the kinetic energy for both cars separately and their combination would give detailed insights into the total energy of motion within this system.