Problem 5 - College Physics

Question

\(\cdot\) The speed of the fastest-pitched baseball was \(45 \mathrm{m} / \mathrm{s},\) and the ball's mass was 145 \(\mathrm{g}\) . (a) What was the magnitude of the mo- mentum of this ball, and how many joules of kinetic energy did it have? (b) How fast would a 57 gram ball have to travel to have the same amount of (i) kinetic energy, and (ii) momentum?

Step-by-Step Solution

Answer

Momentum: 6.525 kg m/s; Kinetic Energy: 146.8125 J; Velocity for same KE: 71.82 m/s; Velocity for same momentum: 114.474 m/s.

Step 1: Convert Mass to Kilograms

First, we need to convert the mass of the 145 g baseball to kilograms.Since 1 gram = 0.001 kilograms, \[ 145 \text{ g} = 0.145 \text{ kg} \]

Step 2: Calculate Momentum

The formula for momentum is \[ p = m \cdot v \]where \(m\) is the mass and \(v\) is the velocity. Substituting the values:\[ p = 0.145 \text{ kg} \times 45 \text{ m/s} = 6.525 \text{ kg m/s} \]

Step 3: Calculate Kinetic Energy

The formula for kinetic energy is \[ KE = \frac{1}{2} m v^2 \]Substitute the given values:\[ KE = \frac{1}{2} \times 0.145 \text{ kg} \times (45 \text{ m/s})^2 = 146.8125 \text{ J} \]

Step 4: Find Velocity for Same Kinetic Energy with New Mass

Given the new mass of 57 grams, which is 0.057 kg, we use the kinetic energy formula:\[ 146.8125 \text{ J} = \frac{1}{2} \times 0.057 \text{ kg} \times v^2 \]Solving for \(v\):\[ v^2 = \frac{146.8125}{0.0285} \]\[ v = \sqrt{5154.8255} = 71.82 \text{ m/s} \]

Step 5: Find Velocity for Same Momentum with New Mass

Using the momentum formula for the new mass:\[ 6.525 \text{ kg m/s} = 0.057 \text{ kg} \times v \]Solve for \(v\):\[ v = \frac{6.525}{0.057} = 114.474 \text{ m/s} \]

Key Concepts

Momentum Calculation

Momentum is a fundamental concept in physics, representing the quantity of motion an object has. It is calculated using the product of an object's mass and its velocity. The formula for momentum ( p ) is given by

• \( p = m \cdot v \)

Here, \( m \) stands for mass (in kilograms) and \( v \) represents velocity (in meters per second). By multiplying these two values, we derive the momentum in units of kilogram meter per second ( kg \, m/s ). Consider a fast-pitched baseball with a mass of 0.145 kg and a speed of 45 m/s. Its momentum is calculated to be 6.525 kg \, m/s , showing how much force the moving baseball would exert upon impact. Remember, momentum depends decorously on both mass and velocity, so any change in these factors directly affects momentum. The importance of momentum extends beyond this single calculation. It's pivotal in understanding collisions, conservation in isolated systems, and the transfer of kinetic energy. Learning to calculate it accurately gives insight into topics ranging from sports science to vehicle safety.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It quantifies how much work an object can perform owing to its velocity. The formula for kinetic energy (\( KE \)) is

• \( KE = \frac{1}{2} m v^2 \)

Where \( m \) is mass in kilograms and \( v \) is velocity in meters per second, this formula helps determine the energy an object carries as it moves. For example, a 0.145 kg baseball traveling at 45 m/s possesses approximately 146.8125 Joules of kinetic energy. This energy reflects the baseball's capacity to perform actions, like breaking a barrier, upon impact. Understanding kinetic energy is crucial in various fields. It's essential for engineers designing vehicles, calculating fuel efficiency, or determining the energy required for different motions in mechanical systems. Mastering this concept can illuminate the relationship between motion and energy, emphasizing the potential impact on and interaction with surrounding environments.

Unit Conversion

Unit conversion is vital to solving physics problems accurately, especially as it ensures dimensional consistency. For example, when dealing with mass in momentum and kinetic energy calculations, converting from grams to kilograms guarantees the results align with the SI units used in other metrics like meters and seconds.Converting mass is straightforward. Knowing that 1 gram equals 0.001 kilograms, we transform masses easily:

• For 145 grams into kilograms, we calculate: \( 145 \, g = 145 \, \times \, 0.001 = 0.145 \, kg \)
• Similarly, a 57-gram mass becomes \( 0.057 \, kg \).

This practice ensures that when performing calculations, each term adheres to standard measurement units, preventing errors that arise from incompatible unit usage. In educational settings, competence with unit conversions fosters critical thinking and versatility in tackling problems across various scientific disciplines. By incorporating these skills, students enhance their ability to achieve precision in problem-solving.

Mass and Velocity Relationship

The relationship between mass and velocity is integral to understanding momentum and kinetic energy. In physics, mass and velocity significantly influence an object's movement, dictating not only how much momentum it carries but also the kinetic energy it possesses.For momentum, the formula \( p = m \cdot v \) shows that increasing mass or velocity directly increases momentum, and vice versa. Thus, heavier or faster objects will have more momentum, a principle observable in phenomena like car crashes, where both speed and vehicle mass result in significant momentum shifts upon impact. Kinetic energy is slightly different. While it also depends on mass and velocity, the formula \( KE = \frac{1}{2} m v^2 \) highlights velocity's squared effect. Thus, despite a smaller mass, drastically increasing velocity can still result in high kinetic energy, explaining why lightweight objects can pack a powerful punch when moving fast.Understanding these relationships can be pivotal in many applications, from designing effective safety gear and automotive components to predicting the behavior of astronomical objects. By mastering both concepts, students can better predict and manipulate outcomes in various mechanical and physical processes.