Problem 7 - College Physics

Question

\(\bullet\) 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) 3p; (b) \(\sqrt{3}p\)

Step 1: Understand the Problem

We are given that a cat has a momentum with magnitude \( p \). We need to find the momentum for a dog that has three times the mass of the cat in two scenarios: when it has the same speed and when it has the same kinetic energy as the cat.

Step 2: Momentum Formulae

Recall that momentum is given by the formula \( p = mv \), where \( m \) is mass and \( v \) is velocity. Also, kinetic energy is given by \( KE = \frac{1}{2}mv^2 \). These will be key in solving the problem.

Step 1: Same Speed

For scenario (a), the speed \( v \) is the same for both the cat and the dog. Since the dog has three times the mass of the cat and shares the same speed, its momentum will be \( 3mv \). Therefore, the momentum of the dog will be \( 3p \).

Step 2: Same Kinetic Energy

For scenario (b), both the cat and the dog have the same kinetic energy. If the kinetic energy of the cat is \( KE = \frac{1}{2}mv^2 \), then for the dog having three times the mass but the same kinetic energy, we use:\[ \frac{1}{2}(3m)v_d^2 = \frac{1}{2}mv^2 \]Solving for \( v_d \), the dog's speed becomes \( v_d = \frac{v}{\sqrt{3}} \). Thus, the momentum of the dog is:\[ p_d = (3m) \cdot \frac{v}{\sqrt{3}} = \sqrt{3} mv = \sqrt{3}p \]

Step 5: Review Answers

From our calculations: (a) When the dog has the same speed as the cat, the momentum is \( 3p \); (b) When the dog has the same kinetic energy as the cat, the momentum is \( \sqrt{3}p \).

Key Concepts

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. It is dependent on two factors: the mass of the object and its velocity. The standard formula used to calculate kinetic energy is \[ KE = \frac{1}{2}mv^2 \]where:

• \( KE \) is the kinetic energy,
• \( m \) represents mass, and
• \( v \) is the velocity of the object.

When analyzing problems involving kinetic energy, like the one with the cat and the dog, it is important to note how changes in mass and velocity affect their kinetic energy. For instance, in our scenario, both the cat and the dog have the same kinetic energy, but the dog has three times the mass of the cat. This implies that to maintain the same kinetic energy, the dog must move at a different velocity from the cat. Understanding how to manipulate the kinetic energy equation is crucial in solving such problems.

Velocity

Velocity is the speed of something in a given direction. It's a vector quantity, meaning it has both magnitude and direction. In physics problems, velocity (\( v \)) often plays a crucial role in determining other physical properties like momentum and kinetic energy.
For the scenario involving the cat and the dog, velocity interfaced with both kinetic energy and momentum equations. Here, when we give the dog the same kinetic energy as the cat, we need to adjust its velocity due to its higher mass. Using the formula for kinetic energy, we can derive that the dog's velocity should be \( v_d = \frac{v}{\sqrt{3}} \) to balance the higher mass.
Understanding changes in velocity is essential when considering changes in motion. The dynamics of velocity alterations give insight into real-world physics situations, importantly illustrating how larger masses require less speed to match the kinetic energy of smaller, faster-moving objects.

Mass

Mass is a fundamental concept in physics that describes the amount of matter in an object. It is often represented by the symbol \( m \) and is typically measured in kilograms (kg). Mass does not change regardless of an object's location and is different from weight, which is affected by gravity.
In the exercise with the cat and the dog, mass is a pivotal factor. The dog's mass is three times that of the cat. This significantly affects both the momentum and kinetic energy when comparing both animals. With an increase in mass at the same velocity, momentum will increase linearly since momentum is the product of mass and velocity.
It is important to grasp how mass interacts with other physical quantities. For any given kinetic energy, an object with larger mass will have a smaller velocity and vice versa for a fixed speed, the momentum increases with mass.

Momentum Formula

Momentum defines the quantity of motion an object has and is an important concept in understanding motion in physics. It is calculated using the formula \[ p = mv \] where:

• \( p \) represents momentum,
• \( m \) is the mass of the object, and
• \( v \) is its velocity.

In the context of the given problem, the momentum of the dog and the cat was analyzed under different conditions. When both had the same speed, the dog’s momentum became \( 3p \) because its mass is three times that of the cat's. However, when they had the same kinetic energy, the dog's velocity was adjusted, resulting in its momentum being \( \sqrt{3}p \).
Understanding and using the momentum formula involves seeing how changes in mass or velocity affect momentum. Real-world applications necessitate that we consider these interdependencies carefully.