Question

A car of mass \(1200 . \mathrm{kg}\), moving with a speed of 72.0 mph on a highway, passes a small SUV with a mass \(1 \frac{1}{2}\) times bigger, moving at \(2 / 3\) the speed of the car. a) What is the ratio of the momentum of the SUV to that of the car? b) What is the ratio of the kinetic energy of the SUV to that of the car?

Short answer

Answer: The ratio of the momentum of the SUV to that of the car is approximately 0.99996. The ratio of the kinetic energy of the SUV to that of the car is approximately 0.665.

Step-by-step solution

Convert the speed of the car to SI units

To make the calculations using the SI system, convert the speed of the car from mph to meters per second (m/s) using the conversion factor: 1 mile = 1609.34 meters, 1 hour = 3600 seconds. Speed of the car in m/s = 72.0 mph * (1609.34 m/mile) * (1 hour/3600 s) = 32.186 m/s

Calculate the mass and speed of the SUV

The mass of the SUV is given to be 1.5 times bigger than the car's mass: Mass of the SUV = 1.5 * 1200 kg = 1800 kg The speed of the SUV is given as 2/3 the speed of the car: Speed of the SUV = (2/3) * 32.186 m/s = 21.457 m/s

Calculate momenta of the car and the SUV

Momentum can be calculated using the formula: Momentum = Mass * Velocity Momentum of the car = 1200 kg * 32.186 m/s = 38623.2 kg m/s Momentum of the SUV = 1800 kg * 21.457 m/s = 38621.6 kg m/s

Calculate the ratio of momenta of the SUV to that of the car

Divide the momentum of the SUV by the momentum of the car: Momentum ratio = Momentum of SUV / Momentum of car = 38621.6 kg m/s / 38623.2 kg m/s = 0.99996

Calculate kinetic energies of the car and the SUV

Kinetic energy can be calculated using the formula: Kinetic energy = (1/2) * Mass * (Velocity^2) Kinetic energy of the car = (1/2) * 1200 kg * (32.186 m/s)^2 = 624162.648 J Kinetic energy of the SUV = (1/2) * 1800 kg * (21.457 m/s)^2 = 415441.306 J

Calculate the ratio of kinetic energies of the SUV to that of the car

Divide the kinetic energy of the SUV by the kinetic energy of the car: Kinetic energy ratio = Kinetic energy of SUV / Kinetic energy of car = 415441.306 J / 624162.648 J = 0.665 Answer: a) The ratio of the momentum of the SUV to that of the car is approximately 0.99996. b) The ratio of the kinetic energy of the SUV to that of the car is approximately 0.665.

Key concepts

Momentum Calculation

Understanding momentum is an essential part of physics problem-solving, especially when it comes to moving objects. Momentum, represented as the product of an object's mass and its velocity, is crucial for predicting how objects will move and interact. The formula for calculating momentum is simple yet powerful: \( p = m \times v \) where \( p \) is momentum, \( m \) is mass and \( v \) is velocity.

In real-world applications, such as traffic collision analysis, knowing an object's momentum helps us to understand the force involved and the potential impacts of the moving objects upon collision. In our exercise, we calculated the momentum of two vehicles and compared their values, revealing the subtleties of momentum's dependence on both mass and velocity. Although the SUV has a greater mass, its slower speed compared to the car yields a momentum that is nearly the same for both.

Kinetic Energy Calculation

While momentum provides one perspective on the motion of an object, kinetic energy adds another layer, focusing on the work needed to bring an object to a certain speed. Kinetic energy is defined as the energy an object possesses due to its motion and is given by the formula: \( KE = \frac{1}{2} m v^2 \) where \( KE \) represents kinetic energy. Unlike momentum, kinetic energy is directly proportional to the square of velocity, which means that small changes in velocity can cause large changes in kinetic energy.

This is evident in the problem we solved, as the kinetic energy of both the car and the SUV did not have the near-equal values seen with momentum. The car's greater speed had a significant impact, resulting in it possessing more kinetic energy than the heavier, yet slower-moving SUV. Thus, kinetic energy teaches us about the potential destructive power in motion, being a key concept for fields ranging from vehicle safety to roller coaster design.

SI Unit Conversion

For clear communication and standardization in physics problem-solving, it is essential to use the International System of Units (SI). These units enable scientists and engineers worldwide to share and compare findings with precision. The SI unit for speed is meters per second (m/s), and for mass, it is kilograms (kg).

When given a problem with measurements in non-SI units, such as miles per hour for speed, a crucial step is to convert these values to SI units before proceeding with calculations. This is the reason for the initial step in our vehicular problem, where the car's speed was converted from mph to m/s. Both momentum and kinetic energy calculations rely on these standard units to ensure their accuracy and relevance.

Physics Problem Solving

Solving physics problems typically involves a systematic approach that includes understanding the concepts involved, identifying the known and unknown variables, and then applying the appropriate equations. A solid comprehension of the units involved and the ability to convert between them, as demonstrated in the SUV and car problem, is critical.

In this case, the process began with converting units to SI standards. We then proceeded to apply formulas for momentum and kinetic energy. Each step was carefully executed, and intermediate results were used to find the final ratios. This step-by-step methodology is not only useful for solving textbook problems but also for troubleshooting complex situations in the real world. Instilling this systematic approach in students can vastly improve their problem-solving skills across various disciplines.