Question

Rank the following objects from highest to lowest in terms of momentum and from highest to lowest in terms of energy. a) an asteroid with mass \(10^{6} \mathrm{~kg}\) and speed \(500 \mathrm{~m} / \mathrm{s}\) b) a high-speed train with a mass of \(180,000 \mathrm{~kg}\) and a speed of \(300 \mathrm{~km} / \mathrm{h}\) c) a 120 -kg linebacker with a speed of \(10 \mathrm{~m} / \mathrm{s}\) d) a \(10-\mathrm{kg}\) cannonball with a speed of \(120 \mathrm{~m} / \mathrm{s}\) e) a proton with a mass of \(2 \cdot 10^{-27} \mathrm{~kg}\) and a speed of \(2 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\)

Short answer

Question: Rank the following objects in order from highest to lowest in terms of their momentum and kinetic energy: an asteroid with mass 10^6 kg and speed 500 m/s, a train with mass 180,000 kg and speed 300 km/h, a linebacker with mass 120 kg and speed 10 m/s, a cannonball with mass 10 kg and speed 120 m/s, and a proton with mass 2 × 10^-27 kg and speed 2 × 10^8 m/s. Answer: In terms of momentum, the ranking is as follows: 1. Asteroid, 2. Train, 3. Linebacker, 4. Cannonball, 5. Proton. In terms of kinetic energy, the ranking is: 1. Asteroid, 2. Train, 3. Cannonball, 4. Linebacker, 5. Proton.

Step-by-step solution

Convert units

First, let's convert the speed of the train from km/h to m/s (1 km = 1000 m, and 1 hour = 3600 s). Speed of the train is given as 300 km/h. (300 km/h) * (1000 m/km) * (1 h/3600 s) = 83.33 m/s Now, all the speeds are in m/s. We can proceed to the next step.

Calculate momentum

We will use the formula for momentum (p = mv) to find the momentum for each object in kg m/s. a) Momentum of the asteroid (10^6 kg) * (500 m/s) = 5 × 10^8 kg m/s b) Momentum of the train (180,000 kg) * (83.33 m/s) = 14,999,400 kg m/s c) Momentum of the linebacker (120 kg) * (10 m/s) = 1,200 kg m/s d) Momentum of the cannonball (10 kg) * (120 m/s) = 1,200 kg m/s e) Momentum of the proton (2 × 10^-27 kg) * (2 × 10^8 m/s) = 4 × 10^-19 kg m/s

Rank objects by momentum

Now, we will rank the objects from highest to lowest in terms of momentum: 1. Asteroid 2. Train 3. Linebacker 4. Cannonball 5. Proton

Calculate kinetic energy

We will use the formula for kinetic energy (KE = 1/2mv^2) to find the kinetic energy for each object in Joules. a) Kinetic energy of the asteroid (1/2) × (10^6 kg) × (500 m/s)^2 = 1.25 × 10^11 J b) Kinetic energy of the train (1/2) × (180,000 kg) × (83.33 m/s)^2 = 6.25 × 10^8 J c) Kinetic energy of the linebacker (1/2) × (120 kg) × (10 m/s)^2 = 6,000 J d) Kinetic energy of the cannonball (1/2) × (10 kg) × (120 m/s)^2 = 72,000 J e) Kinetic energy of the proton (1/2) × (2 × 10^-27 kg) × (2 × 10^8 m/s)^2 = 4 × 10^-11 J

Rank objects by kinetic energy

Now, we will rank the objects from highest to lowest in terms of kinetic energy: 1. Asteroid 2. Train 3. Cannonball 4. Linebacker 5. Proton So, the final ranking for momentum and kinetic energy are: Momentum: 1. Asteroid, 2. Train, 3. Linebacker, 4. Cannonball, 5. Proton Kinetic energy: 1. Asteroid, 2. Train, 3. Cannonball, 4. Linebacker, 5. Proton

Key concepts

Momentum Calculation

When it comes to understanding the movement of objects, momentum is a crucial concept. It's defined as the product of an object's mass and velocity, expressed by the equation
\[\begin{equation}p = mv,ewline\end{equation}\] where p represents momentum, m is mass in kilograms, and v is velocity in meters per second (m/s). The resulting unit for momentum is kilogram meter per second (kg m/s).
In our example, once we calculate the momentum for each object using this formula, we can easily compare them. Larger momentum signifies that the object will be harder to stop, implying it has more 'oomph' as it moves. It's also a vector quantity, which means it has both magnitude and direction, although in this exercise, we focus on magnitude.
Here's a practical exercise tip: always double-check if your units are consistent (like converting all velocities to m/s) before comparing momentums, as this avoids any mix-up that could lead to incorrect rankings or comparisons.

Kinetic Energy Calculation

While momentum gives us a sense of the 'drive' an object has due to its motion, kinetic energy helps us understand how much work an object can do by virtue of its motion. Kinetic energy (KE) is calculated with the formula
\[\begin{equation}KE = \frac{1}{2}mv^2,\end{equation}\] where KE is kinetic energy in joules (J), m is the object's mass in kilograms, and v is its velocity in meters per second (m/s). Unlike momentum, kinetic energy is a scalar quantity—it only has magnitude and no direction.
In our example, when we calculated the KE for each object, it allowed us to determine which has the most potential to do work. Remember, when comparing kinetic energies, recognize that velocity plays a more significant role due to it being squared in the equation. This can drastically affect the ranking of objects, especially when their velocities are notably different.

Unit Conversion

Unit conversion is pivotal in physics to ensure all quantities are measured in compatible units before performing calculations. As seen in the solution for this problem, we converted the train's speed from kilometers per hour (km/h) to meters per second (m/s). Knowing that 1 km equals 1000 meters and 1 hour is 3600 seconds, we use the conversion factors to change units:
\[\begin{equation}(300 \frac{\text{km}}{\text{h}}) \times (1000 \frac{\text{m}}{\text{km}}) \times (\frac{1}{3600} \frac{\text{h}}{\text{s}}) = 83.33 \frac{\text{m}}{\text{s}}.\end{equation}\] Mastery of unit conversion is a must-have skill for accurately comparing physical quantities. It's what ensures coherent conclusions in a physics problem and avoiding this step can lead to significant mistakes. Always include the necessary conversion before proceeding to other calculations.

Comparing Physical Quantities

In physics problems, especially when we need to rank or arrange objects in a certain order like highest to lowest, comparing physical quantities is crucial. Once we have the right units and correct calculations for momentum and kinetic energy, as we did in the textbook solution, the comparison becomes straightforward.
It's important to note that while momentum and kinetic energy both describe aspects of an object's motion, they are different quantities and tell us different things about the object. Momentum relates to how much force is needed to change the object’s movement, whereas kinetic energy relates to the work needed to bring it to a stop.
A useful hint for students would be to look at the magnitudes of each physical quantity and remember that for kinetic energy, due to the square of velocity, even a small change in speed may have a significant impact on the energy an object has.