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Chapter 2: Problem 188

A particle is projected vertically upwards with velocity $30 \mathrm{~ms}^{-1}$. Find the ratio of average speed and instantaneous velocity after 6 s. \(\left[\mathrm{g}=10 \mathrm{~ms}^{-1}\right]\) (A) \((1 / 2)\) (B) 2 (C) 3 (D) 4

Short Answer

Expert verified
Based on the step-by-step solution, the short answer is as follows: The ratio of average speed and instantaneous velocity after 6 seconds is \(0/(-30) = 0\). However, there might be an error in the problem's statement or the answer choices, as 0 is not among the given options.

Step by step solution

01

Write down the given values and equations

Given Initial velocity, u = 30 m/s Acceleration, a = -10 m/s^2 (since acceleration due to gravity is downward) Time, t = 6 s We need to find the ratio: (Average speed)/(Instantaneous velocity)
02

Calculate the final velocity

To find the final velocity after t=6s, we can use the following equation: v = u + a*t Substituting the values, we get: v = 30 + (-10)*6 v = 30 - 60 = -30 m/s (The negative sign indicates that the particle is moving downward after 6 seconds)
03

Calculate the average speed

The average speed is defined as the total distance covered divided by the total time taken. Since the particle is projected upwards and then falls back to its highest point, from where it starts to fall back, we should take into account only the upward distance traveled during the 6 seconds interval. We can use the following equation for distance (s): s = u*t + 0.5*a*t^2 Substituting the values, we get: s = 30*6 - 0.5*10*(6^2) s = 180 - 180 = 0 Since the distance is 0, the average speed is also 0.
04

Calculate the instantaneous velocity

We have already found the final velocity in step 2, which is the instantaneous velocity after 6 seconds. So, v_inst = -30 m/s
05

Find the ratio of the average speed and the instantaneous velocity

Now, we can calculate the ratio between the average speed and the instantaneous velocity: (Average speed)/(Instantaneous velocity) = 0/(-30) = 0 But the given options do not have 0 as an option, so there might be an error in the problem's statement or the answer choices.

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