Problem 9 - An Introduction to Physical Science

Question

An airplane flying directly eastward at a constant speed travels \(300 \mathrm{km}\) in \(2.0 \mathrm{h}\). (a) What is the average velocity of the plane? (b) What is its instantaneous velocity?

Step-by-Step Solution

Answer

(a) 150 km/h. (b) 150 km/h eastward.

Step 1: Define Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. Here, the airplane is traveling directly east, so the displacement is the same as the distance. Use the formula \( v_{avg} = \frac{\text{displacement}}{\text{time}} \).

Step 2: Calculate Average Velocity

Displacement \( = 300 \, \text{km} \) and time \( = 2.0 \, \text{hours} \). Substitute these values into the formula for average velocity: \[ v_{avg} = \frac{300 \, \text{km}}{2.0 \, \text{h}} = 150 \, \text{km/h}.\]

Step 3: Define Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a particular moment in time. For an object moving at a constant speed in a straight line, the instantaneous velocity is the same as the velocity at any given moment, which is equal to the average velocity.

Step 4: Determine Instantaneous Velocity

Since the airplane is traveling at a constant speed and direction, the instantaneous velocity is equal to the average velocity computed earlier, which is \( 150 \, \text{km/h} \) eastward.

Key Concepts

Instantaneous Velocity

When we talk about instantaneous velocity, we mean the speed and direction of an object at a specific point in time. To put it simply, imagine taking a snapshot of an object's movement. The velocity you capture in that instant is the instantaneous velocity.
This concept is particularly important when movements are not at a steady pace. However, in scenarios like our airplane flying at a constant speed and in a straight path, the instantaneous velocity remains the same as the average velocity.
Here's why:

• Constant Speed: The airplane's speed does not change over the period of travel.
• Straight Line Motion: The direction remains eastward, just like the average trajectory.

Generally, if there's no change in speed and direction, the instantaneous velocity and average velocity will be equal, making our airplane's velocity straightforward to determine.

Displacement

Displacement refers to the change in position of an object in a specific direction. It is a vector quantity, meaning it has both magnitude and direction.
While discussing our airplane example, the displacement is fairly easy to figure out since the flyer's travel is directly eastward.
Key aspects of displacement:

• Direction: In our case, eastward, matching the path the airplane takes.
• Magnitude: This is the straight-line distance from the starting position to the final location. For the airplane, it's the whole 300 km.

This is not just about how far you have traveled but also where you traveled. It simplifies to distance when the direction remains unchanged, as with our airplane example.

Constant Speed

Constant speed is when an object covers equal distances in equal intervals of time, without any variation in speed. This concept is pivotal for determining both average and instantaneous velocity.
For the airplane flying east:

• The airplane uniformly covers the full journey of 300 km in 2 hours.
• The distance covered in any smaller time segment remains proportional and consistent.

A constant speed implies that the forces acting on the airplane have reached equilibrium, allowing for a smooth flight that maintains a consistent motion without fluctuations.

Time

Time is a fundamental scalar quantity that helps measure the duration of events. In our example, time is crucial for calculating both the average and instantaneous speeds.
Considering the pilot's 2-hour journey to cover 300 km:

• Time provides the interval to observe the airplane's journey.
• By dividing the total displacement by this time, we determine the average velocity, which in this case, also represents the instantaneous velocity.

Time, measured in consistent units such as hours, minutes, or seconds, gives us a lens through which to view velocity and speed measurements effectively, ensuring precision in such calculations.