Question

A man standing on the edge of a cliff throws a stone straight up with initial speed $u$ and then throws another stone straight down with an equal initial speed u from the similar position. Find the ratio of speeds, if the stones would have attained when they hit the ground at the base of the cliff?
$\begin{align}
  & A.2:1 \\
 & B.1:2 \\
 & C.1:1 \\
 & D.3:1 \\
\end{align}$

Answer 1

Hint: First of all find out the initial velocity as well as final velocity, when the stone has been thrown upwards and then downwards. When the stone is thrown upwards, the final velocity will be zero and the direction of acceleration will be negative. In the downward motion of stone, $u$ is zero as it will be the final velocity in this case. These all will help us in calculating the answer.

Complete answer:
Speed of a body or the magnitude of velocity is the same at similar height. Therefore, when the man throws the stone with speed $u$, it will come back to the person with the similar speed $u$.
Therefore we can write that,
$v=u+at$
Here the acceleration of the motion will be the acceleration due to gravity which will be negative. Hence we can write that,

$\begin{align}
  & v=u-gt \\
 & \because v=0 \\
 & 0=u-gt \\
\end{align}$

Then the value of initial velocity $u$ will be
$u=gt$

Here the acceleration is negative due its direction of motion.
Now let us look when the stone comes down,
While it is coming down,
$v=0-gt$

Here also the acceleration is negative as the motion is in downward direction.
Therefore we can write that,
$v=gt$
From the equations we got,
$u=gt$
$v=gt$
We can conclude that,
$\left| v \right|=\left| u \right|$
So when a man throws a stone upward with a speed of $u$ and downward with speed $u$, the initial velocity from the cliff will be equal. Therefore this may result in similar final velocity.
Hence the ratio will be $1:1$

So, the correct answer is “Option C”.

Note:
Gravitational acceleration is given as the free fall acceleration of a body in vacuum without any drag. This is the steady gain in speed as a result of only the force of gravitational attraction. This is a vector quantity also.