Question

A 1200-kg SUV is moving along a straight highway at 12.0 m/s. Another car, with mass 1800 kg and speed 20.0 m/s, has its center of mass 40.0 m ahead of the center of mass of the SUV (\(\textbf{Fig. E8.54}\)). Find (a) the position of the center of mass of the system consisting of the two cars; (b) the magnitude of the system's total momentum, by using the given data; (c) the speed of the system's center of mass; (d) the system's total momentum, by using the speed of the center of mass. Compare your result with that of part (b).

Short answer

(a) Center of mass is 24 m from SUV. (b) Total momentum is 50400 kg m/s. (c) Speed of center of mass is 16.8 m/s. (d) Total momentum verification: 50400 kg m/s, consistent with (b).

Step-by-step solution

Identify essential formulas for center of mass

To find the center of mass of the system, we use the formula for the center of mass of a system of particles: \( x_{cm} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2} \), where \( m_1 \) and \( m_2 \) are the masses of the two cars, and \( x_1 \) and \( x_2 \) are their respective positions.

Calculate the position of the center of mass

Using the given data, set \( x_1 = 0 \) (position of SUV) and \( x_2 = 40 \) m (position of the other car relative to SUV). The masses are \( m_1 = 1200 \) kg and \( m_2 = 1800 \) kg.\[x_{cm} = \frac{1200 \times 0 + 1800 \times 40}{1200 + 1800} = \frac{72000}{3000} = 24 \text{ m from SUV}\]

Determine the formulas for total momentum and speed of center of mass

The total momentum of a system is given by \( p = m_1v_1 + m_2v_2 \). The speed of the center of mass is \( v_{cm} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \).

Calculate the total momentum of the system

Using the given velocities \( v_1 = 12 \text{ m/s} \) for the SUV and \( v_2 = 20 \text{ m/s} \) for the other car:\[p = 1200 \times 12 + 1800 \times 20 = 14400 + 36000 = 50400 \text{ kg m/s}\]

Calculate the speed of the center of mass

Using the formula for the speed of the center of mass:\[v_{cm} = \frac{1200 \times 12 + 1800 \times 20}{1200 + 1800} = \frac{50400}{3000} = 16.8 \text{ m/s}\]

Verify total momentum using speed of center of mass

Calculate the total momentum again using the speed of the center of mass:\[p = (1200 + 1800) \times 16.8 = 3000 \times 16.8 = 50400 \text{ kg m/s}\]The result matches with the total momentum calculated earlier, confirming our calculations.

Key concepts

Momentum and Its Importance

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. For any object, it is calculated as the product of its mass and velocity:

• Momentum (\( p \) ) = Mass (\( m \) ) x Velocity (\( v \) )

It is a vector quantity, meaning it has both magnitude and direction. In a system of particles, like the two cars on the highway, the total momentum is the sum of the individual momenta of each object. Calculating the total momentum is essential for understanding the system's dynamics and conserving momentum in interactions.
For our exercise, the momentum of each car contributes to the total system momentum, which is crucial when analyzing collisions or interactions in mechanics.

Understanding a System of Particles

A system of particles refers to a group of two or more objects that interact or are examined collectively. In the context of physics, especially involving center of mass calculations, systems of particles are studied to understand the collective behavior of these bodies. The center of mass is a critical concept here, acting as an average position weighed by mass within the system.
For our problem with two cars, treating them as a system allows for the calculation of a combined center of mass. This system-based approach provides insights into how different factors like mass and velocity affect the entire system. Moreover, it simplifies the representation of dynamics, such as how the system as a whole will respond to forces.

Defining and Calculating Velocity

Velocity is a measure of how fast an object changes its position, a fundamental concept in kinematics. Unlike speed, which is scalar, velocity is a vector, indicating both the rate of change of position and the direction of motion. To determine how systems behave, calculating the velocity of the center of mass of a system of particles is crucial.
The velocity of the center of mass helps us track the motion of the entire system as a single point. For our exercise, determining the velocity of the system's center of mass involves using the weighted average of each component's velocity. This approach provides a simplified view of the system's overall motion.

Step-by-Step Position Calculation

The position calculation of the center of mass involves determining its location based on the positions and masses of the objects in the system. For the cars in our problem, we start by acknowledging each mass and its respective position. To find the center of mass, we use:

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

Here, each position is weighted by its mass, which means more massive objects heavily influence the center of mass’s position. This calculation gives a point that effectively balances the two cars' positions like a seesaw. It is a crucial step for understanding the system's overall behavior and predicting how it will continue in motion when external forces act upon it.