Question

A block moves in a straight line with velocity $ v $ for time $ {t_0} $ .Then, its velocity becomes $ 2v $ for next $ {t_0} $ time. Finally, its velocity becomes $ 3v $ for time $ T $ . If average velocity during the complete journey was $ 2.5v $ , then find $ T $ in terms of $ {t_0} $ .

Answer 1

Hint : In this solution, we will use the relation of the average velocity of the object as the ratio of the total distance travelled by the objected to the total amount of time taken by the object to cover that distance. By using this relation, we’ll be able to find $ T $ in terms of $ {t_0} $ .

Formula used: In this solution we will be using the following formula,
  $ {\text{Avg}}{\text{. speed = }}\dfrac{{{\text{Total}}\,{\text{distance}}}}{{{\text{Total}}\,{\text{time}}}} $

Complete step by step answer
We’ve been given that a block moves in a straight line with velocity $ v $ for time $ {t_0} $ .Then, its velocity becomes $ 2v $ for next $ {t_0} $ time. Finally, its velocity becomes $ 3v $ for time $ T $ . To calculate the average velocity of this block, we need to find the total distance traveled by this block. For the time when the block is moving is with a velocity $ v $ , the distance it will travel in time $ {t_0} $ will be
 $ {d_1} = v{t_0} $
For the next time bracket, when the object is moving with velocity $ 2v $ , the distance it will cover in $ {t_0} $ will be
 $ {d_2} = 2v{t_0} $
Similarly, for the last part of its motion, when its velocity is $ 3v $ , the distance it will cover in time $ T $ will be
 $ {d_3} = 3vT $
Then the average velocity of the object can be calculated as the ratio of the distance of the object to the total time taken by the object to travel this distance as:
 $ {v_{avg}} = \dfrac{{{d_1} + {d_2} + {d_3}}}{{{t_0} + {t_0} + T}} $
On placing the values of $ {d_1},{d_2},{d_3} $ , we get,
 $ {v_{avg}} = \dfrac{{v{t_0} + 2v{t_0} + 3vT}}{{2{t_0} + T}} $
 $ \Rightarrow {v_{avg}} = \dfrac{{3v{t_0} + 3vT}}{{2{t_0} + T}} $
But we’ve been given the average velocity in the question as $ 2.5v $ , so we can say that
 $ 2.5v = \dfrac{{3v{t_0} + 3vT}}{{2{t_0} + T}} $
Dividing both sides by $ v $ , we get,
 $ 2.5 = \dfrac{{3{t_0} + 3T}}{{2{t_0} + T}} $
 $ \Rightarrow 5{t_0} + 2.5T = 3{t_0} + 3T $
Solving for $ T $ , we get
 $ T = 4{t_0} $.

Note
Here, we can use this simple relation for average velocity only because the object is moving in a straight line otherwise, we would have to take into account the direction of the object. In such cases, it is always beneficial to find the total distance travelled by the object as the object is moving different distances with different velocities so we can use the total distance it travels to calculate the average velocity.