Question

Rohit completes a semicircular path of radius \(\mathrm{R}\) in 10 seconds. Calculate average speed and average velocity in \(\mathrm{ms}^{-1}\) (A) \([(2 \pi R) / 10],(2 R / 10)\) (B) \((\pi R / 10),(\mathrm{R} / 10)\) (C) \((\pi R / 10),(2 R / 10)\) (D) \([(2 \pi R) / 10],(\mathrm{R} / 10)\)

Short answer

The short answer is: (C) \((\pi R / 10),(2 R / 10)\).

Step-by-step solution

Find Distance Covered by Rohit

The distance is the total length of the semicircular path Rohit travels. The formula for the circumference of a circle is \(2\pi{R}\), so for a semicircle, the distance would be half of this, which is $$ D = (\pi R). $$

Find Displacement of Rohit

Displacement is the straight distance from Rohit's starting point to his endpoint. Rohit begins at one end of the semicircle and takes a straight line to the other end, which is equivalent to the diameter of the semicircular path. The formula for diameter is 2R. Thus, the displacement is $$ s = 2R. $$

Calculate Average Speed

To find the average speed, we need to divide the total distance traveled by the time taken. In this case, the time taken is given as 10 seconds. Therefore, the average speed would be represented as $$ v_{avg(speed)} = \frac{D}{t} = \frac{\pi R}{10}. $$

Calculate Average Velocity

To find the average velocity, we need to divide the total displacement by the time taken. In this case, the time taken is also given as 10 seconds. Therefore, the average velocity would be represented as $$ v_{avg(velocity)} = \frac{s}{t} = \frac{2R}{10}. $$

Compare Answers to Given Options

Comparing our calculated average speed and average velocity to the given options, we get: (A) \([(2 \pi R) / 10],(2 R / 10)\) (B) \((\pi R / 10),(\mathrm{R} / 10)\) (C) \((\pi R / 10),(2 R / 10)\) (D) \([(2 \pi R) / 10],(\mathrm{R} / 10)\) So, the correct answer to the problem is (C) \((\pi R / 10),(2 R / 10)\).

Key concepts

Semicircular Path

When analyzing Rohit's journey, it's crucial to understand what a "semicircular path" implies. Imagine a circle, and then cut it into two equal halves—each half forms a semicircle. Rohit moves along one of these halves, following the curved path rather than cutting straight across. The curve represents half the total circumference of the circle.
To calculate the length of this path, we use the formula for the circumference of a full circle, which is given by:

• Full circle circumference: \(2\pi R\)

This formula is based on the radius \(R\) of the circle. Since Rohit only travels half of the circle, the distance he covers is:

• Length of semicircle: \(\pi R\)

This semicircular path is pivotal as it establishes the distance regarding the total path covered.

Distance and Displacement

Distance and displacement, though related, are two distinct concepts in physics. Understanding these differences is key for calculating average speed and velocity.

• Distance: This refers to the total journey length covered, irrespective of direction. In Rohit's scenario, the distance he walks is the length of the semicircular path, which calculates to \(\pi R\).
• Displacement: This is the direct straight-line distance from the starting to endpoint. While the path Rohit takes is curved, his displacement is a straight line across the circle's diameter. This line is equivalent to twice the radius of the circle, so the displacement is \(2R\).

Distance is always equal to or greater than the displacement, which highlights the story of Rohit's journey around the circle versus across it.

Time Calculation

In our exercise, Rohit's journey along the semicircular path takes a precise duration of 10 seconds. For any movement, time is a crucial factor as it allows us to determine rates of change, such as speed and velocity.
When a problem explicitly provides a time duration, such as the 10 seconds here, it simplifies calculations significantly. This duration (10 seconds) becomes the denominator in the formulas used to calculate both average speed and average velocity. Hence, understanding and correctly utilizing time data is critical for determining motion parameters. It's a straightforward yet essential element that can't be overlooked in calculations.

Motion Equations

Motion equations assist in defining the average speed and velocity. Let's break them down:
Average Speed: Average speed is defined as the total distance traveled divided by the total time taken. It's a scalar quantity, meaning it only considers magnitude not direction:

• \(v_{avg(speed)} = \frac{\text{Distance}}{\text{Time}} = \frac{\pi R}{10}\)

This shows that regardless of direction, it's the path length over the time that matters.
Average Velocity: Average velocity considers both magnitude and direction, hence it's a vector quantity. It is calculated as the displacement divided by the time taken:

• \(v_{avg(velocity)} = \frac{\text{Displacement}}{\text{Time}} = \frac{2R}{10}\)

This highlights the divergence between speed and velocity as average velocity accounts for the direct line displacement and not the curved path.
These motion formulas allow us to derive meaningful insights from Rohit's movement, shedding light on how different paths and directions affect the outcome.