Problem 7 - College Physics

Question

The magnitude of the momentum of a cat is \(p\). What would be the magnitude of the momentum (in terms of \(p\) ) of a dog having three times the mass of the cat if it had (a) the same speed as the cat, and (b) the same kinetic energy as the cat?

Step-by-Step Solution

Answer

(a) Momentum is \(3p\); (b) Momentum is \(\sqrt{3}p\).

Step 1: Understanding the Problem

We need to find the momentum of a dog in terms of the momentum of a cat, given two different scenarios. First, if the dog has three times the mass of the cat and moves at the same speed. Second, if both the dog and the cat have the same kinetic energy.

Step 1: Define Variables

Let's define \( m_c \) as the mass of the cat,\( v \) as the velocity (speed) of both the cat and the dog. The dog's mass \( m_d = 3m_c \). The momentum of the cat is given as \( p = m_c v \).

Step 2: Calculating Momentum (Same Speed)

For a dog with the same speed as the cat, the momentum is determined using the formula \( p = mv \). Substitute the dog's mass and the speed:\[ p_d = (3m_c) v = 3(m_c v) = 3p.\]Therefore, the momentum of the dog, when it moves at the same speed as the cat, is \( 3p \).

Step 3: Understanding Kinetic Energy

Kinetic energy is calculated using the formula \( KE = \frac{1}{2}mv^2 \).Given that the kinetic energies of the dog and cat are equal, we can find the dog's speed by equating their kinetic energies and solve for the momentum.

Step 4: Calculating Velocity for Same KE

Set the kinetic energies equal:\[\frac{1}{2}m_c v^2 = \frac{1}{2}(3m_c)v_d^2,\]Solve for \(v_d^2\):\[v_d^2 = \frac{v^2}{3} \rightarrow v_d = \frac{v}{ ext{sqrt}(3)}.\]Thus, the dog's speed is \( \frac{v}{ ext{sqrt}(3)} \).

Step 5: Calculating Momentum (Same KE)

Using the speed from the previous step, calculate the dog's momentum:\[ p_d = (3m_c) v_d = 3m_c \left(\frac{v}{\sqrt{3}}\right) = \sqrt{3} m_c v = \sqrt{3} p.\]Thus, the momentum of the dog, when it has the same kinetic energy as the cat, is \( \sqrt{3} p \).

Step 7: Conclusion

For part (a), the dog's momentum is \( 3p \). For part (b), the dog's momentum is \( \sqrt{3}p \), given their different conditions of speed and kinetic energy.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It's an important concept in physics because it helps us understand how objects move and interact with each other. The formula to calculate kinetic energy is \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. This equation tells us that kinetic energy is directly proportional to the mass of the object and the square of its velocity.
For example, if two objects move at the same speed but one has a greater mass, the heavier one will have more kinetic energy. Similarly, a small increase in velocity leads to a much larger increase in kinetic energy because velocity is squared in the equation. Understanding kinetic energy is crucial for solving problems involving the movement of objects, such as the exercise with the cat and dog.

Mass

Mass is a measure of how much matter an object has and plays a crucial role in physics. It's often confused with weight, but unlike weight, mass does not change based on location. Mass is important in calculating both momentum and kinetic energy.
In the exercise, the dog's mass is three times that of the cat's. If the masses are \( m_c \) for the cat and \( m_d = 3m_c \) for the dog, we see how mass affects momentum and kinetic energy. With more mass, the dog has more momentum at the same speed and different kinetic energy characteristics as well.

• Bigger mass = more momentum for the same speed.
• Mass is constant irrespective of location.
• Affects both momentum and kinetic energy calculations.

This concept allows us to predict how different mass distributions change the motion and energy requirements of objects.

Velocity

Velocity, often referred to simply as 'speed' in everyday language, is a vector quantity. This means it has both magnitude and direction. It's key to understanding how momentum and kinetic energy behave. Formally, velocity is the change in position per unit time, and it plays an integral role in physics equations.
For instance, the momentum \( p \) is calculated using the formula \( p = mv \), where \( m \) is mass and \( v \) is velocity. This indicates that an increase or decrease in velocity directly affects the amount of momentum an object has.
In the exercise, when the dog and cat have the same kinetic energy, the dog's new velocity must be calculated as \( \frac{v}{\sqrt{3}} \) to match this energy despite having a larger mass. Understanding velocity helps in predicting how objects will move under different conditions.

Physics Equations

Physics equations form the backbone of describing physical phenomena with precision. They are mathematical representations that help us predict and understand the behavior of objects in the universe.

• Momentum equation: \( p = mv \)
• Kinetic energy equation: \( KE = \frac{1}{2} mv^2 \)

These equations allow us to solve for unknown variables when given enough information, as seen in the exercise where we calculated the dog's momentum in two scenarios. Physics equations require a clear understanding of each involved variable, such as mass, velocity, and kinetic energy. By mastering these equations, students can efficiently analyze problems involving moving objects, making it easier to resolve complex questions like comparing the momentum of a cat and a dog under different conditions. Understanding and applying these principles is essential for success in physics.