Problem 3 - University Physics with Modern Physics

Question

( a) Show that the kinetic energy \(K\) and the momentum magnitude \(p\) of a particle with mass \(m\) are related by \(K=p^{2} / 2 m\) (b) \(A 0.040\) -kg cardinal (Richmondena cardinalis) and a 0.145 -kg baseball have the same kinetic energy. Which has the greater magnitude of momentum? What is the ratio of the cardinal's magnitude of momentum to the baseball's? (c) A \(700-\mathrm{N}\) man and a \(450-\mathrm{N}\) woman have the same momentum. Who has the greater kinetic energy? What is the ratio of the man's kinetic energy to that of the woman?

Step-by-Step Solution

Answer

(a)\( K = \frac{p^2}{2m}\); (b) Baseball has greater momentum; ratio is 0.524. (c) Woman has greater kinetic energy; ratio is 0.643.

Step 1: Kinetic Energy and Momentum Relationship

First, recall the formula for kinetic energy, given by \( K = \frac{1}{2} m v^2 \), where \( v \) is the velocity of the particle. Also, recall the formula for momentum, given by \( p = mv \). To find the relationship between \( K \) and \( p \), substitute \( v = \frac{p}{m} \) back into the kinetic energy equation: \( K = \frac{1}{2} m \left( \frac{p}{m} \right)^2 \). Simplifying yields \( K = \frac{p^2}{2m} \).

Step 2: Momentum Comparison for Bird and Baseball

Using the kinetic energy-momentum relationship, both the cardinal and the baseball have the same kinetic energy \( K \), so \( \frac{p_{bird}^2}{2m_{bird}} = \frac{p_{baseball}^2}{2m_{baseball}} \). Solving for the momentum gives \( p_{bird} = \sqrt{2m_{bird}K} \) and \( p_{baseball} = \sqrt{2m_{baseball}K} \). Since the mass of the bird is smaller than the baseball's mass, for the same kinetic energy, \( p_{bird} < p_{baseball} \), thus the baseball has greater momentum. The ratio of momenta is \( \frac{p_{bird}}{p_{baseball}} = \sqrt{\frac{m_{bird}}{m_{baseball}}} \). Plugging in the given masses, \( \frac{p_{bird}}{p_{baseball}} = \sqrt{\frac{0.040}{0.145}} \approx 0.524 \).

Step 3: Kinetic Energy Comparison for Man and Woman

The man and the woman have equal momentum, \( p_{man} = p_{woman} \), implying \( m_{man}v_{man} = m_{woman}v_{woman} \). However, kinetic energy depends on both velocity and mass, \( K=\frac{p^2}{2m} \). Substituting their weights (where weight, \( W = mg \), and thus \( m = \frac{W}{g} \)), the kinetic energy for the man is \( K_{man} = \frac{p^2}{2(\frac{700}{9.8})} \), and for the woman, \( K_{woman} = \frac{p^2}{2(\frac{450}{9.8})} \). Simplifying gives \( K_{man} < K_{woman} \) because the man's mass is greater, so the woman has greater kinetic energy. The ratio is \( \frac{K_{man}}{K_{woman}} = \frac{450}{700} \approx 0.643 \).

Key Concepts

Physics problem solving

Solving physics problems involves comprehending relationships between different physical quantities. In the case of kinetic energy and momentum, understanding how these concepts interrelate can aid in solving problems efficiently. To start, one needs to know the basic formulas: kinetic energy is given by \( K = \frac{1}{2} m v^2 \) and momentum by \( p = mv \).

When asked to express kinetic energy in terms of momentum, it's crucial to recognize the commonality they share through the variable \( v \) (velocity). By substituting \( v = \frac{p}{m} \) in the kinetic energy formula, the relationship \( K = \frac{p^2}{2m} \) is derived.

Understanding such derivations is key to tackling physics problems effectively. It provides a deep insight into how different quantities are interdependent, making problem-solving more intuitive.

Mass and momentum relationship

Mass and momentum are closely tied in physics. Momentum, being the product of mass and velocity \( (p = mv) \), signifies how much motion an object possesses.

In problems involving comparison, like the cardinal and baseball scenario, knowing the impact mass has on momentum is crucial. Given equal kinetic energy for both objects, the larger mass - the baseball in this case - tends to have greater momentum if kinetic energy \( K \) remains unchanged. This stems from the relationship \( p = \sqrt{2mK} \).

The ratio of their momenta can be determined by the masses \( \frac{p_{bird}}{p_{baseball}} = \sqrt{\frac{m_{bird}}{m_{baseball}}} \). This relationship highlights the role mass plays in determining momentum magnitude when other energy criteria are constant. Recognizing these relationships helps in predicting outcomes in various physical contexts.

Kinetic energy comparison

Kinetic energy depends on both mass and velocity. This means even with equal momentum, two objects, like the man and woman in the scenario, can have different kinetic energy.

The key formula to remember here is \( K = \frac{p^2}{2m} \). Substituting known quantities, such as mass being weight over gravity \( m = \frac{W}{g} \), allows us to calculate and compare kinetic energy effectively. Here, despite having equal momentum, the man's greater mass results in lesser kinetic energy than the woman.

By assessing the ratio \( \frac{K_{man}}{K_{woman}} = \frac{450}{700} \), students can visualize the effect of mass on energy, reinforcing their understanding of kinetic energy nuances. This aids in grasping how momentum translates into kinetic energy differences under identical momentum conditions.