A boy aims a gun at a bird from a point, at a horizontal distance of 100 m. If the gun can impart a velocity of 500 ms⁻¹ to the bullet. At what height above the bird must he aim his gun in order to hit it (take g = 10 ms⁻²)

The correct option/answer is 20 cm.

Let us break down the problem step-by-step to understand how to find the height above the target at which the gun must be aimed.
When a bullet is fired from a gun, it follows a parabolic trajectory due to the acceleration caused by gravity ($g$) acting vertically downwards. If the boy aims directly at the bird, the bullet will fall below the bird by the time it travels the horizontal distance. To compensate for this downward fall, he must aim at a height $h$ above the bird.

Let:
- Horizontal distance to the bird, $x=100\text{m}$
- Initial velocity of the bullet, $u=500{\text{ms}}^{-1}$
- Acceleration due to gravity, $g=10{\text{ms}}^{-2}$

Step 1: Calculate the time taken by the bullet to travel the horizontal distance
Since there is no horizontal acceleration (neglecting air resistance), the horizontal velocity remains constant at $u=500{\text{ms}}^{-1}$.
The time $t$ taken to cover the horizontal distance $x$ is given by:
$t=\frac{x}{u}$
Substituting the given values:
$t=\frac{100}{500}=0.2\text{s}$

Step 2: Calculate the vertical displacement (fall) during this time
Initially, when aimed horizontally or nearly horizontally over this short time, the initial vertical velocity component is zero. The vertical distance $h$ by which the bullet falls under gravity in time $t$ is given by the equation of motion:
$h=\frac{1}{2}g{t}^2$
Substituting the values of $g$ and $t$:
$h=\frac{1}{2}\times 10\times {\left(0.2\right)}^2$
$h=5\times 0.04=0.2\text{m}$

Step 3: Convert the height into centimeters
Since the options are provided in centimeters, we convert the result:
$h=0.2\text{m}=0.2\times 100\text{cm}=20\text{cm}$

Therefore, the boy must aim his gun at a height of 20 cm above the bird in order to hit it.