Question

An object is thrown vertically upwards to a height of 10 m. Calculate its
(a) Velocity with which it was thrown upward.
(b) Time taken by the object to reach the highest point.

Answer 1

Hint: At the maximum height, the velocity of the object becomes zero. Use the kinematic equation which does not contain a time variable to determine the initial velocity of the object. To determine the time taken by the object to reach the highest point, use the kinematic equation which does not have the displacement variable.

Formula used:
Kinematic equation, \[{v^2} = {u^2} + 2as\]
Here, v is the final velocity, u is the initial velocity, a is the acceleration and s is the displacement.
\[v = u + at\]
Here, t is the time.

Complete step by step answer:
We have given the maximum vertical height attained by the object is \[h = 10\,m\].
(a)To calculate the initial velocity of the object with which it is thrown upward, we can use kinematic equation as follows,
\[{v^2} = {u^2} - 2gh\]
Here, v is the final velocity, u is the initial velocity, g is the acceleration due to gravity and h is the height attained by the object.
We know that at the maximum height, the velocity of the object becomes zero. Therefore, we can substitute 0 for v in the above equation.
\[{0^2} = {u^2} - 2gh\]
\[ \Rightarrow {u^2} = 2gh\]
\[ \Rightarrow u = \sqrt {2gh} \]
Substituting \[9.8\,m/{s^2}\] for g and 10 m for h in the above equation, we get,
\[u = \sqrt {2\left( {9.8} \right)\left( {10} \right)} \]
\[ \Rightarrow u = 14\,m/s\]
Therefore, the initial velocity of the object is 14 m/s.

(b) To calculate the time taken by the object to reach the height h, we can use following kinematic equation,
\[v = u - gt\]
Since final velocity v is zero, the above equation becomes,
\[u = gt\]
\[ \Rightarrow t = \dfrac{u}{g}\]
Substituting 14 m/s for u and \[9.8\,m/{s^2}\] for g in the above equation, we get,
\[t = \dfrac{{14}}{{9.8}}\]
\[ \Rightarrow t = 1.43\,{\text{s}}\]
Therefore, the time taken by the object to reach the highest point is 1.43 seconds.

Note:
Students should always remember that for the vertically upward motion of the body, the sign of the acceleration due to gravity must be negative and that for the downward motion, the sign of the acceleration due to gravity must be positive. To solve these types of questions, students should remember all the kinematic equations.