Question

A Plymouth with a mass of \(2210 \mathrm{~kg}\) is moving along a straight stretch of road at \(105 \mathrm{~km} / \mathrm{h}\). It is followed by a Ford with mass \(2080 \mathrm{~kg}\) moving at \(43.5 \mathrm{~km} / \mathrm{h}\). How fast is the center of mass of the two cars moving?

Short answer

The speed of the center of mass of the two cars is \(20.74 \mathrm{~m/s}\).

Step-by-step solution

Conversion of Units

Convert the speeds of both the Plymouth and the Ford from km/h to m/s. This is done by multiplying the given value by \(1000 \mathrm{m}\) (the number of meters in a kilometer) and then dividing by \(3600 \mathrm{s}\) (the number of seconds in an hour). Thus, the speeds are: Plymouth - \(105 \mathrm{~km/h} \times \frac{1000 \mathrm{~m}}{3600 \mathrm{~s}} = 29.17 \mathrm{~m/s}\) and Ford - \(43.5 \mathrm{~km/h} \times \frac{1000 \mathrm{~m}}{3600 \mathrm{~s}} = 12.08 \mathrm{~m/s}\).

Calculation of Center of Mass Speed

Use the formula for the velocity of the center of mass, which is \[ V_{cm} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \] where \(m_1\) and \(v_1\) are the mass and velocity of the Plymouth respectively, and \(m_2\) and \(v_2\) are the mass and velocity of the Ford respectively. Plugging the given values into the formula gives \[\frac{(2210 \mathrm{~kg} \times 29.17 \mathrm{~m/s}) + (2080 \mathrm{~kg} \times 12.08 \mathrm{~m/s})}{2210 \mathrm{~kg} + 2080 \mathrm{~kg}} = 20.74 \mathrm{~m/s}\]. Hence, the center of mass of the two cars is moving with a speed of \(20.74 \mathrm{~m/s}\).

Key concepts

Conversion of Units

When working through physics problems, it's often necessary to convert between units of measurement. This is because equations are typically defined within a specific unit system, and using consistent units is essential to obtaining accurate results. In the given exercise, the speeds of two cars are initially provided in kilometers per hour (km/h). This is a common unit for speed but not the standard unit in physics.

To align with the International System of Units (SI), these speeds need to be converted to meters per second (m/s). The process involves two simple steps: multiplying by the distance in meters within one kilometer (1 km = 1000 m), and then dividing by the number of seconds in one hour (1 h = 3600 s). By doing so, we can ensure that our velocity calculations are properly formatted for use in physics equations.

Velocity Calculation

Velocity is the rate at which an object changes its position, and it's a vector quantity, meaning it has both magnitude and direction. Calculating velocity is a foundational aspect of kinematics, a branch of mechanics within physics. In terms of our problem, once the units are converted, we can proceed to calculate the velocity of the center of mass (V_cm) for the two cars.

The formula \[ V_{cm} = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \] combines the masses and velocities of both cars. It essentially averages the velocities, weighted by the mass of each car. Thus, a heavier car's velocity contributes more to the center of mass velocity than a lighter car. Thanks to the previous conversion of units, we can accurately determine that the center of mass for the two vehicles moves at 20.74 m/s. This velocity is crucial in understanding kinetic energies, momenta, and dynamic behaviors of systems.

Physics Problem-Solving

Solving physics problems often requires a systematic approach: understanding the problem, identifying what you know and what you need to find out, and finally, selecting and applying the relevant formulas. These steps help in solving complex problems and achieving the correct answers.

The key in physics problem-solving is to work methodically. Start with a clear understanding of the principles involved, in this case, the center of mass and velocity. Next, write down all given data and convert necessary units before substituting them into the appropriate formulas. This structured approach not only aids in solving the problem at hand but also enhances your ability to tackle similar problems in the future. Remember to also check the units of your final answer to ensure they are sensible within the context of the question.