Question

A particle is thrown in upward direction with initial velocity of \(60 \mathrm{~m} / \mathrm{s}\). Find average speed and average velocity after 10 seconds. \(\left[\mathrm{g}=10 \mathrm{~ms}^{-2}\right]\) (A) \(26 \mathrm{~ms}^{-1}, 16 \mathrm{~ms}^{-1}\) (B) \(26 \mathrm{~ms}^{-1}, 10 \mathrm{~ms}^{-1}\) (C) \(20 \mathrm{~ms}^{-1}, 10 \mathrm{~ms}^{-1}\) (D) \(15 \mathrm{~ms}^{-1}, 25 \mathrm{~ms}^{-1}\)

Short answer

The average speed and average velocity after 10 seconds are \(20 \mathrm{~ms}^{-1}\) and \(10 \mathrm{~ms}^{-1}\), respectively. So, the correct option is (C).

Step-by-step solution

Calculate the final position after 10 seconds

To find the final position (s) after 10 seconds, we need to use the following equation of motion: \(s = ut + \frac{1}{2}at^2\) where u is the initial velocity (60 m/s), t is the time (10 seconds), and a is the acceleration (-10 m/s^2, negative because gravity acts downward) Putting the values, we get: \(s = (60)(10) - \frac{1}{2}(10)(10)^2\) \(s = 600 - 500\) \(s = 100 m\)

Calculate the final velocity after 10 seconds

To find the final velocity (v) after 10 seconds, we'll use the following equation of motion: \(v = u + at\) where u is the initial velocity (60 m/s), t is the time (10 seconds), and a is the acceleration (-10 m/s^2) Putting the values, we get: \(v = 60 - (10)(10)\) \(v = 60 - 100\) \(v = -40 m/s\)

Calculate the total distance

To calculate the total distance traveled by the particle, we need to consider both the upward and downward paths. The downward path will have the same distance as the upward path. Therefore, the total distance will be: \(Total Distance = 2 \times s = 2 \times 100\) \(Total Distance = 200 m\)

Calculate the average speed

The average speed is the total distance traveled divided by the total time. In this case, the total distance is 200 m and the total time is 10 seconds: \(Average Speed = \frac{Total Distance}{Total Time} = \frac{200}{10}\) \(Average Speed = 20 m/s\)

Calculate the average velocity

The average velocity is the change in displacement divided by the total time. The displacement after 10 seconds is 100 m (final position) with upward direction: \(Average Velocity = \frac{Displacement}{Total Time} = \frac{100}{10}\) \(Average Velocity = 10 m/s\) This implies that the correct option is (C) \(20 \mathrm{~ms}^{-1}, 10 \mathrm{~ms}^{-1}\).

Key concepts

Average Speed

Average speed is a basic yet essential concept in physics, particularly within the context of motion. It is defined as the total distance traveled by an object divided by the total time taken to cover that distance. This measure gives us an idea of how fast something is moving, regardless of the direction.

In the given exercise, the particle travels upwards and then downwards. Both paths are considered in calculating total distance. Here is why the complete journey is important:

• The particle first moves upwards, slowing down due to gravity until it momentarily stops at its highest point.
• Then it moves back downwards, covering the same path.

Thus, the average speed considers this total distance. Hence, in our scenario, after 10 seconds, the total distance becomes 200 meters, making the average speed calculation:
\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{200}{10} = 20 \text{ m/s}\]
Calculating average speed demands you focus on overall travel, not just where the object ended up.

Average Velocity

Average velocity is defined differently from average speed because it considers the direction of movement, not just the magnitude. It is calculated as the change in displacement (the straight-line distance from the starting point to the endpoint) divided by the total time.

In physics, it is crucial to distinguish between distance and displacement.

• Distance covers the entire path taken, while
  
• Displacement the shortest path between the start and the endpoint.

In this exercise:

• The initial and final positions of the particle allow us to find the displacement, which is 100 meters (upward).
• After 10 seconds, the average velocity becomes:
  \[\text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}} = \frac{100}{10} = 10 \text{ m/s}\]

Note that this varies from average speed because it directly relates to what happened in a straight line from start to finish, taking direction into account.

Equations of Motion

The equations of motion form a foundational aspect of classical mechanics, allowing us to predict how objects move under various forces. They are advantageous in scenarios involving constant acceleration, like our projectile exercise, influenced by gravity.

These equations express relationships between displacement, velocity, time, and acceleration. Here's a quick recap on what's relevant to our particle scenario.

• The first equation \(v = u + at\) enables us to calculate the particle's final velocity by considering its initial velocity, acceleration due to gravity, and time.
• The second equation \(s = ut + \frac{1}{2}at^2\) helps calculate the displacement after a specific time.

The exercise uses both equations:

• To calculate the displacement (100 meters) after 10 seconds, showing its peak position in an upward path.
• To determine the final velocity at 10 seconds (-40 m/s), indicating the particle's downward motion as it returns.

Despite seeming complex, these equations simplify our understanding of how different variables interact in motion scenarios, thereby enhancing our problem-solving capability in physics.