A ball is thrown vertically upwards with a velocity $\text{'}u\text{'}$. The velocity with which it falls to the earth again is:

Step 1: Given Data

Initial velocity for upward motion $=u$

The final velocity for upward motion be $v=0$.

Initial velocity for the downward motion $u=0$

The final velocity for downward motion be $v$.

Let the acceleration due to gravity be $g$.

Let the time taken be $t$.

Step 2: Time Taken for Upward Motion

For the vertically upward motion $g$ is negative as it opposes the motion.

According to Newton's first law of motion

$v=u-gt$

$\Rightarrow 0=u-gt$

$\Rightarrow t=\frac{u}{g}$ $\left(1\right)$

Step 3: Time Taken for Downpward Motion

For vertically downward motion $g$ is positive as it is in the direction of motion.

According to Newton's first law of motion

$v=u+gt$

$\Rightarrow v=0+gt$

$\Rightarrow t=\frac{v}{g}$ $\left(2\right)$

Step 4: Calculate Final velocity

From equation $\left(1\right)$ and $\left(2\right)$ we get

$t=\frac{u}{g}=\frac{v}{g}$

$\Rightarrow v=u$

Hence, the velocity with which the ball falls on the ground is $u$.