Question

Running along a rectangular track \(50 \mathrm{~m} \times 40 \mathrm{~m}\) you complete one lap in \(100 \mathrm{~s}\). What is your average velocity for the lap?

Short answer

Answer: The average velocity of the runner is 0 m/s.

Step-by-step solution

Determine the track's dimensions

The given dimensions of the rectangular track are \(50 \mathrm{~m}\) and \(40 \mathrm{~m}\).

Calculate the total distance traveled

To find the total distance traveled, we need to find the perimeter of the rectangular track. The formula for the perimeter of a rectangle is \(P = 2 \times (L+W)\), where L is the length and W is the width. Here, the length L is \(50 \mathrm{~m}\) and the width W is \(40 \mathrm{~m}\). So, the perimeter or total distance is: \(P = 2 \times (50 + 40) = 2 \times 90 = 180 \mathrm{~m}\)

Determine the time taken to complete one lap

The given time taken to complete one lap is \(100 \mathrm{~s}\).

Find the displacement after completing one lap

Since the runner starts and ends at the same point after completing one lap, the total displacement is \(0 \mathrm{~m}\).

Calculate the average velocity

The formula for average velocity is: \(Average \ Velocity = \cfrac{Total \ Displacement}{Total \ Time}\) As the displacement is \(0 \mathrm{~m}\), the average velocity is: \(Average \ Velocity = \cfrac{0 \mathrm{~m}}{100 \mathrm{~s}} = 0 \cfrac{\mathrm{m}}{\mathrm{s}}\) Hence, the average velocity for the lap is \(0 \cfrac{\mathrm{m}}{\mathrm{s}}\).

Key concepts

rectangular track

When you think of a rectangular track, picture something similar to a long rectangle made up of two sets of parallel sides. Specifically, this shape has two longer sides, which we refer to as the length, and two shorter sides known as the width. So when someone mentions a rectangular track, they are simply referring to a running path shaped like a rectangle.

In the case where our track dimensions are 50 meters by 40 meters, these dimensions dictate the total layout of the running path. This kind of track is commonly used in athletics because it provides a clear path while ensuring that the runners maintain a consistent rhythm throughout their lap travels. Understanding the dimensions of a rectangular track is fundamental in solving problems related to motion on such paths.

perimeter calculation

To solve problems involving motion around a rectangular track, one essential concept is perimeter calculation. The perimeter of a rectangle is the total distance around the track—a key component for tasks involved with running laps. We calculate it using the formula:

• \( P = 2 \times (L + W) \)

Here, \(L\) is the length and \(W\) is the width of the rectangular track.

Applying this to our track with dimensions 50 meters and 40 meters, we plug in these numbers:

• \( P = 2 \times (50 + 40) = 2 \times 90 = 180 \text{ meters} \)

This 180 meters is how far one would travel in one complete lap, which is crucial when calculating other elements like average velocity or total time taken.

displacement in motion

Displacement in motion refers to the change in position of an object from its starting point to its ending point. It's important to note that displacement considers the shortest path between two points, not the path traveled.

When a runner completes a full lap around a rectangular track, they return to their starting point. Therefore, in terms of physics, the displacement is zero because there is no net change in position. It's the same as thinking about a round trip in your car; even if you've driven 100 kilometers, if you start and end at your home, your displacement is zero.

• Displacement concerns only net position change.
• For closed tracks, like a rectangular path, displacement after one full loop is zero.

time and distance relationship

Time and distance are central themes in understanding motion, especially when dealing with tracks. In our exercise, the time taken to complete one lap is given as 100 seconds. Coupled with the perimeter of the track which is 180 meters, we can understand concepts like speed and velocity.

Speed is calculated as:

• \( \text{Speed} = \frac{\text{Total Distance}}{\text{Time}} = \frac{180\text{ m}}{100\text{ s}} = 1.8\ \text{m/s}\)

This represents how fast the runner covers the entire distance of the track.

On the other hand, average velocity is calculated based on displacement over time. Since the displacement is zero in our scenario (returning to the starting point), the average velocity turns out to be zero.

• This illustrates that while speed might be constant, velocity can be zero if you end where you started.
• Understanding the relationship between distance, time, and how they define speed and velocity is critical in analyzing motion-related scenarios on such tracks.