Question

A particle moves \(4 \mathrm{~m}\) in the south direction. Then it moves \(3 \mathrm{~m}\) in the west direction. The time taken by the particle is 2 second. What is the ratio between average speed and average velocity? (A) \((5 / 7)\) (B) \((7 / 5)\) (C) \((14 / 5)\) (D) \((5 / 14)\)

Short answer

The ratio between average speed and average velocity is (B) \((7 / 5)\).

Step-by-step solution

Calculate the total distance traveled

In the problem, the particle moves 4m in south direction and 3m in west direction. The total distance traveled by the particle is the sum of these distances: Total distance = 4m (south) + 3m (west) = 7m

Calculate the total displacement of the particle

The displacement of the particle is the shortest distance between the initial and final positions of the particle. We can treat the problem as a right-angled triangle with sides 4m and 3m. The displacement is the hypotenuse of the triangle. Using the Pythagorean theorem, we can calculate the displacement: Displacement = \(\sqrt{(4m)^2 + (3m)^2}\) = \(\sqrt{16m^2 + 9m^2}\) = \(\sqrt{25m^2}\) = 5m

Calculate the average speed

The average speed is calculated by dividing the total distance by the time taken. In this case, the total distance is 7m, and the time taken is 2 seconds: Average speed = Total distance / Time = 7m / 2s = 3.5 m/s

Calculate the average velocity

The average velocity is calculated by dividing the total displacement by the time taken. In this case, the total displacement is 5m, and the time taken is 2 seconds: Average velocity = Total displacement / Time = 5m / 2s = 2.5 m/s

Find the ratio between average speed and average velocity

To find the ratio between the average speed and average velocity, divide the average speed by the average velocity: Ratio = Average speed / Average velocity = 3.5 m/s / 2.5 m/s = 7/5 The answer is (B) \((7 / 5)\).

Key concepts

Average Speed

Average speed is a measure of how fast an object is moving irrespective of direction. It is calculated by taking the total distance traveled and dividing it by the total time taken. For the particle in our exercise, it moved a total distance of 7 meters in 2 seconds.

To find the average speed:

• Total distance = 7 meters
• Total time = 2 seconds
• Average speed = Total distance / Total time = 7m / 2s = 3.5 m/s

By understanding average speed, we know the rate at which the particle covers ground over the entire journey.

Displacement

Displacement refers to the shortest distance between the starting and ending point and includes direction. In this exercise, the particle created a right-angled path by first moving south and then moving west.

The displacement is the hypotenuse in this right-angled scenario. Using understanding from geometry and physics, displacement is generally less than or equal to the total distance, unless all movement happens in a straight line.

In this case, by applying the Pythagorean theorem, it is:

• Displacement = \( \sqrt{(4m)^2 + (3m)^2} \) = \( \sqrt{16m^2 + 9m^2} \) = \( \sqrt{25m^2} \) = 5 meters

This gives us both the magnitude and direction of the shortest path from start to finish.

Total Distance

The total distance is simply the full path length that the particle has traveled, regardless of direction. In our exercise, the particle moved first south and then west, giving us segments of the path to add together.

• South: 4 meters
• West: 3 meters
• Total distance = 4m + 3m = 7 meters

Unlike displacement, total distance considers every turn and twist in the journey, not focusing on just the start and end points.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, applied here to find displacement. In any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This theorem is essential to calculating indirect distances like displacement.

In our problem, the particle’s movement results in a right triangle with legs:

• One leg: 4 meters
• Other leg: 3 meters

Displacement (hypotenuse) is calculated using:

• \( Displacement = \sqrt{(4m)^2 + (3m)^2} \)

The Pythagorean theorem simplifies understanding spatial relationships, especially when direct measurement isn’t available.