Problem 3 - College Physics

Question

Find the magnitude of the linear momentum of (a) a \(7.1-\mathrm{kg}\) bowling ball traveling at \(12 \mathrm{~m} / \mathrm{s}\) and \((\mathrm{b})\) a \(1200-\mathrm{kg}\) automobile traveling at \(90 \mathrm{~km} / \mathrm{h}\).

Step-by-Step Solution

Answer

Bowling Ball: 85.2 kg·m/s; Automobile: 30000 kg·m/s.

Step 1: Understanding Linear Momentum

Linear momentum is the product of an object's mass and velocity. It is given by the formula: \( p = mv \), where \( p \) is the linear momentum, \( m \) is the mass, and \( v \) is the velocity.

Step 1: Convert Units for the Automobile

Before calculating the momentum for the automobile, convert its velocity from kilometers per hour to meters per second. Using the conversion: \(1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s}\), we get: \[ v = 90 \, \text{km/h} = 90 \times \frac{1}{3.6} \, \text{m/s} \approx 25 \, \text{m/s} \]

Step 2: Calculate Momentum for the Bowling Ball

Use the momentum formula for the bowling ball: \( m = 7.1 \, \text{kg} \), \( v = 12 \, \text{m/s} \) Calculate: \[ p = mv = 7.1 \, \text{kg} \times 12 \, \text{m/s} = 85.2 \, \text{kg} \cdot \text{m/s} \]

Step 3: Calculate Momentum for the Automobile

Use the momentum formula for the automobile with the converted velocity: \( m = 1200 \, \text{kg} \), \( v = 25 \, \text{m/s} \) (after conversion) Calculate: \[ p = mv = 1200 \, \text{kg} \times 25 \, \text{m/s} = 30000 \, \text{kg} \cdot \text{m/s} \]

Step 5: Conclude With Results

The linear momentum of the bowling ball is \(85.2 \, \text{kg} \cdot \text{m/s}\), and the linear momentum of the automobile is \(30000 \, \text{kg} \cdot \text{m/s}\).

Key Concepts

Mass and Velocity

To understand linear momentum, we first need to understand its components: mass and velocity. Mass refers to how much matter an object contains, often measured in kilograms. It determines how much an object resists acceleration when a force is applied. Velocity, on the other hand, refers to the speed of the object in a specific direction, typically measured in meters per second (m/s). Velocity is not just about how fast something moves; it's also about the direction it's moving in. Together, these two factors form the core of linear momentum.

• Mass - A scalar quantity indicating the amount of matter.
• Velocity - A vector quantity indicating speed and direction.

Imagine a heavy object like a bowling ball or a light object like a feather. Both have mass, but the bowling ball’s larger mass means it carries more momentum at the same speed. Similarly, if two identical cars travel at different speeds, the faster car has more momentum because of its higher velocity.

Unit Conversion

Before calculating linear momentum, it's essential to have consistent measurement units. Often, we need to convert units to ensure they align properly for accurate calculations. This is known as unit conversion. For instance, you might have measured velocity in kilometers per hour (km/h) but need to convert this to meters per second (m/s) for consistency with the International System of Units (SI).
The conversion factor from kilometers per hour to meters per second is:

• To convert, multiply the velocity by \( \frac{1}{3.6} \). So, if you have a velocity of 90 km/h:
  \[ v = 90 \times \frac{1}{3.6} \approx 25 \, \text{m/s} \]

This conversion is critical in problems where accurate linear momentum calculations are necessary. Always check the units first before proceeding with calculations.

Momentum Calculation

With mass and velocity understood and units correctly converted, you're ready to calculate momentum. Linear momentum is calculated using the formula:
\[ p = mv \]Where \(p\) represents momentum, \(m\) is mass, and \(v\) is velocity.

• For a bowling ball with mass \(7.1 \, \text{kg}\) and velocity \(12 \, \text{m/s}\):
  \[ p = 7.1 \, \text{kg} \times 12 \, \text{m/s} = 85.2 \, \text{kg} \cdot \text{m/s} \]
• For an automobile weighing \(1200 \, \text{kg}\) moving at \(25 \, \text{m/s}\) (after converting from 90 km/h):
  \[ p = 1200 \, \text{kg} \times 25 \, \text{m/s} = 30000 \, \text{kg} \cdot \text{m/s} \]

This straightforward calculation allows us to understand and compare the momentum of different objects, showcasing the relationship between mass, velocity, and the concept of inertia.