Problem 98 - College Physics

Question

\(\bullet\) \(\bullet\) The aircraft carrier \(U S S\) George Washington has mass \(1.0 \times 10^{8} \mathrm{kg} .\) When its engines are developing their full power of \(260,000\) hp, the George Washington travels at its top speed of 35 knots \((65 \mathrm{km} / \mathrm{h}) .\) If 70\(\%\) of the power output of the engines is applied to pushing the ship through the water, what is the magnitude of the force of water resistance that opposes the carrier's motion at this speed?

Step-by-Step Solution

Answer

The force of water resistance opposing the carrier is approximately 7,518,544 N.

Step 1: Convert Power Units

First, we need to convert the power from horsepower to watts. We know that \[1 ext{ hp} = 746 ext{ watts}.\] Thus, the total power in watts is:\[260,000 ext{ hp} \times 746 ext{ watts/hp} = 193,960,000 ext{ watts}.\] Since only 70% of this power is applied to pushing the ship, the effective power is:\[0.70 \times 193,960,000 ext{ watts} = 135,772,000 ext{ watts}.\]

Step 2: Convert Speed Units

Convert the speed of the aircraft carrier from knots to meters per second. We know:\[35 ext{ knots} = 65 ext{ km/h}.\]Convert kilometers per hour to meters per second using the conversion factor:\[ \frac{1 ext{ km/h}}{3.6} = ext{ m/s}.\]Thus, the speed is:\[\frac{65}{3.6} \approx 18.06 ext{ m/s}.\]

Step 3: Calculate Force of Water Resistance

Using the relation between power, force, and velocity:\[P = F \cdot v,\]where \(P\) is the effective power and \(v\) is the velocity, we solve for \(F\) (the force of water resistance):\[F = \frac{P}{v} = \frac{135,772,000 ext{ watts}}{18.06 ext{ m/s}} \approx 7,518,544 ext{ N}.\] This is the magnitude of the force of water resistance that opposes the carrier's motion.

Key Concepts

Power conversion

When dealing with Physics problems, understanding the conversion of power units can be crucial. Power, in this context, refers to the rate at which work is done or energy is transferred. It is often measured in watts in scientific applications but in horsepower (hp) in other contexts like automotive and marine situations. To convert horsepower to watts, remember the conversion factor:

• 1 hp = 746 watts.

So, if we have a power of 260,000 hp, multiply it by 746 to get the power in watts.
That gives us:

• 260,000 hp \( \times \) 746 W/hp = 193,960,000 watts.

Understanding this conversion allows us to apply power more effectively in calculations related to forces and movement.

Water resistance force

Water resistance, or drag force, is a critical factor when looking at objects moving through water, such as a ship. This force acts opposite to the direction of motion, essentially slowing the object down. It's vital to understand that the effective power applied by any marine engine has to overcome this resistance to maintain a certain speed.In our case, only 70% of the aircraft carrier's engine power is used to push it through the water. This "applied power" is what counters the water resistance. To evaluate water resistance:

• Calculate the effective power available by multiplying total power by the percentage used: \( 0.70 \times 193,960,000 \text{ watts} = 135,772,000 \text{ watts} \).

The magnitude of water resistance can be determined with the relationship between power, force, and velocity, as it indicates how much force is needed to overcome the drag at a given speed.

Unit conversion

Unit conversion is a fundamental skill in Physics, allowing you to express measurements in a way that is most useful for solving problems. For speed, you'll often need to convert from one unit to another. Here, the speed of the aircraft carrier is given in both knots and kilometers per hour (km/h), but we usually need it in meters per second (m/s) for calculations involving force and power.Follow these steps for conversion:

• 1 knot is approximately equal to 1.852 km/h.
• The conversion from km/h to m/s is achieved via division by 3.6.

For the given speed:

• 65 km/h \( \div \) 3.6 \( \approx \) 18.06 m/s.

Understanding unit conversions ensures you get the correct figures when doing calculations and helps maintain consistency across different measurement systems.

Force calculation

To figure out the force of water resistance, you rely on the fundamental equation that ties power, force, and velocity together: \( P = F \cdot v \). Here, \( P \) is the effective power in watts, \( F \) is the force in newtons, and \( v \) is the speed in meters per second.To solve for force:

• Rearrange the formula to \( F = \frac{P}{v} \).
• Insert the effective power and speed values.

Given:

• \( P = 135,772,000 \text{ watts} \)
• \( v = 18.06 \text{ m/s} \)

Calculate \( F \):

• \( F = \frac{135,772,000}{18.06} \approx 7,518,544 \text{ newtons} \).

This force represents the power needed to overcome the resistance of water, highlighting the energy expenditure required for motion.

Speed conversion

Converting speed from one unit to another is a frequent task in Physics, especially in problems involving motion. Speed can be represented in many units, each suited for different types of analysis or applications.This exercise involves:

• Converting knots to kilometers per hour and then to meters per second.

Initially, we know that 35 knots equals 65 km/h.
To proceed to meters per second, use the suitable conversion:

• \( \frac{65}{3.6} \approx 18.06 \text{ m/s} \).

Correct speed conversion is vital as it ensures the calculations for other parameters like force are accurate. It gives you a common ground (m/s in this instance) to apply equations and derive solutions.