Question

The muzzle velocity of a projectile fired from a \(9 \mathrm{~mm}\) handgun is \(1200 \mathrm{ft} / \mathrm{s}\). (a) How many miles per hour is this? (b) How many meters per second?

Short answer

(a) 822.86 mph; (b) 365.76 m/s.

Step-by-step solution

Convert Feet per Second to Miles per Hour

First, we need to convert the muzzle velocity from feet per second (ft/s) to miles per hour (mph). The conversion factor between these units is 1 mile = 5280 feet and 1 hour = 3600 seconds. The formula for the conversion is:\[\text{Miles per hour} = \text{Feet per second} \times \frac{3600 \text{ seconds/hour}}{5280 \text{ feet/mile}}.\]Substitute the given velocity:\[1200 \times \frac{3600}{5280} = \text{822.86 mph (approximately)}.\]

Convert Feet per Second to Meters per Second

Next, we will convert the velocity from feet per second to meters per second (m/s). The conversion factor is 1 foot = 0.3048 meters. Use the formula:\[\text{Meters per second} = \text{Feet per second} \times 0.3048.\]Substituting the given velocity:\[1200 \times 0.3048 = \text{365.76 m/s (approximately)}.\]

Key concepts

Velocity Conversion

Understanding velocity conversion is crucial when dealing with different units of speed and distance measurements. When you have a velocity given in one unit, such as feet per second, you might need to convert it into another unit, like miles per hour or meters per second. This is often necessary in physics problems or real-world applications like projectile motion analysis for consistency and clarity in measurements.

Converting velocity involves using specific conversion factors that bridge two different units. This process ensures that you're accurately translating the speed without losing the physical meaning of the measurement. Velocity conversions are common in fields such as engineering, physics, and even automotive industries. Each application may require different units based on the object's speed or the accepted measurement standards in that field.

Feet per Second to Miles per Hour

Converting feet per second (ft/s) to miles per hour (mph) might seem complex at first, but it's quite straightforward with a proper understanding of the conversion factors.

Remember:

• 1 mile is equivalent to 5280 feet
• 1 hour contains 3600 seconds

With these factors, you can use the formula: \[ \text{Miles per hour} = \text{Feet per second} \times \frac{3600}{5280} \] This formula allows you to convert any velocity in ft/s to mph by plugging in the value. In the example of a handgun muzzle velocity at 1200 ft/s, substituting in the formula shows that it equals approximately 822.86 mph.

This conversion is particularly useful when expressing velocities suitable for longer distances and times, like in transportation or aviation.

Feet per Second to Meters per Second

The conversion from feet per second (ft/s) to meters per second (m/s) is another fundamental conversion in scientific and engineering tasks.

The conversion factor used here comes from the metric system:

• 1 foot is equivalent to 0.3048 meters

The calculation formula derives directly from this: \[ \text{Meters per second} = \text{Feet per second} \times 0.3048 \] By inserting the muzzle velocity of 1200 ft/s into this equation, the result is approximately 365.76 m/s.

This conversion is often more applicable in scientific contexts where metric measurements are preferred, especially in international settings or when working with formulas in physics.

Conversion Factors

Conversion factors serve as the bridges between different measurement systems, allowing us to translate one unit into another accurately. These factors are fixed values, such as the number of feet in a mile or the number of meters in a foot, and are essential for maintaining consistency across measurements.

To successfully convert between units, especially for velocity, always:

• Determine the units you need to convert from and to
• Use the appropriate conversion factor
• Set up your equation correctly, remembering to multiply or divide as necessary

In our context of a projectile's velocity, identifying and applying these conversion factors for both feet to miles and feet to meters ensures units are consistent and make sense within the physical context of the problem.

Projectile Motion

Projectile motion refers to the motion of an object projected into the air, subject only to gravity and air resistance. Understanding the initial velocity is key in predicting the path, distance, and time of flight.

An example of this would be calculating the muzzle velocity of a bullet. Knowing its velocity in accurate units helps predict how far and fast it will travel under given conditions.

Factors to consider in projectile motion analysis include:

• Initial speed or velocity
• Angle of projection
• Effects of gravity, which typically accelerates down at \(9.8 \text{ m/s}^2\)

Unit conversions, such as from feet per second to meters per second or miles per hour, play a critical role in ensuring that velocity is measured accurately, enabling precise calculations in predicting projectile paths or comparing the speeds of different projectiles.