Problem 4 - College Physics

Question

\(\cdot \mathrm{A} 1200 \mathrm{kg}\) cur is moving on the freeway at 65 \(\mathrm{mph}\) (a) Find the magnitude of its momentum and its kinetic energy in SI units. (b) If a 2400 \(\mathrm{kg}\) SUV has the same speed as the 1200 \(\mathrm{kg}\) car, how much momentum and kinetic energy does it have?

Step-by-Step Solution

Answer

The car's momentum is 34,872 kg m/s and kinetic energy is 507,602.88 J. The SUV's momentum is 69,744 kg m/s and kinetic energy is 1,015,205.76 J.

Step 1: Convert Speed to SI Unit

The speed of the car and the SUV is given as 65 mph. First, convert this speed to meters per second (m/s). We know that 1 mile is 1609.34 meters and 1 hour is 3600 seconds. Therefore,\[\mathrm{Speed} = 65 \text{ mph} = \frac{65 \times 1609.34}{3600} \text{ m/s} = 29.06 \text{ m/s}\].

Step 2: Calculate Momentum for the Car

Momentum \( p \) is given by \( p = mv \), where \( m \) is the mass and \( v \) is the velocity. For the car,\[ p = 1200 \times 29.06 = 34872 \text{ kg m/s}\].

Step 3: Calculate Kinetic Energy for the Car

Kinetic energy \( KE \) is given by \( KE = \frac{1}{2} mv^2 \). For the car,\[ KE = \frac{1}{2} \times 1200 \times (29.06)^2 = 507,602.88 \text{ J} \].

Step 4: Calculate Momentum for the SUV

For the SUV, the momentum \( p \) is calculated the same way: \( p = mv \).\[ p = 2400 \times 29.06 = 69744 \text{ kg m/s} \].

Step 5: Calculate Kinetic Energy for the SUV

For the SUV, the kinetic energy is calculated the same way: \( KE = \frac{1}{2} mv^2 \).\[ KE = \frac{1}{2} \times 2400 \times (29.06)^2 = 1,015,205.76 \text{ J} \].

Key Concepts

SI Units Conversion

When working with problems in physics, especially those involving motion, it is crucial to use the International System of Units (SI). This ensures consistency and simplicity in calculations. To convert speed from miles per hour (mph) to meters per second (m/s), which is the necessary SI unit, you use the conversion factors:

• 1 mile = 1609.34 meters
• 1 hour = 3600 seconds

For instance, if you have a speed of 65 mph, you can convert it by multiplying by the number of meters in a mile and then dividing by the number of seconds in an hour. So,the speed conversion is \[65 \text{ mph} = \frac{65 \times 1609.34}{3600} \text{ m/s} = 29.06 \text{ m/s}\]This process ensures all calculations are in the correct unit system, allowing the physics formulas to be applied directly.

Momentum Calculation

Momentum is a measure of how much motion an object possesses and is a vital concept in physics. It can be calculated using the formula:\[ p = mv \]where:

• \( p \) is the momentum.
• \( m \) is the mass of the object in kilograms (kg).
• \( v \) is the velocity of the object in meters per second (m/s).

For a car of mass 1200 kg traveling at 29.06 m/s, the momentum is calculated as:\[ p = 1200 \times 29.06 = 34872 \text{ kg m/s} \]For a 2400 kg SUV at the same velocity, the momentum is:\[ p = 2400 \times 29.06 = 69744 \text{ kg m/s} \]Understanding momentum helps explain how the mass and velocity of an object affect its resistance to changes in motion.

Kinetic Energy Formula

Kinetic energy (KE) is the energy an object possesses due to its motion. The formula to calculate kinetic energy is:\[ KE = \frac{1}{2} mv^2 \]where:

• \( KE \) is the kinetic energy measured in joules (J).
• \( m \) is the mass in kilograms (kg).
• \( v \) is the velocity in meters per second (m/s).

For a car with a mass of 1200 kg moving at 29.06 m/s, the kinetic energy is:\[ KE = \frac{1}{2} \times 1200 \times (29.06)^2 = 507,602.88 \text{ J} \]Similarly, for a 2400 kg SUV at the same velocity, the kinetic energy is:\[ KE = \frac{1}{2} \times 2400 \times (29.06)^2 = 1,015,205.76 \text{ J} \]Kinetic energy provides insight into how much work an object can perform due to its motion.

Mass and Velocity Relationship

The mass and velocity of an object are crucial factors that influence both momentum and kinetic energy. To better understand this relationship, consider the following: - **Momentum**: Since momentum is calculated by multiplying mass and velocity, an increase in either will increase momentum proportionally. For example, doubling the mass or velocity will double the momentum. - **Kinetic Energy**: Kinetic energy depends on the square of the velocity; thus, a larger impact comes from changes in velocity compared to changes in mass. For instance, doubling the velocity will quadruple the kinetic energy, while doubling the mass will only double the kinetic energy. These relationships show why a heavier or faster-moving object is more difficult to stop or change direction. Understanding the balance between mass and velocity is key to solving dynamic physics problems effectively.