Question

A gun is aimed at a horizontal target. It takes 1/2 s for the bullet to reach the target. The bullet hits the target x metre below the aim. Then, x is equal to
A. $\dfrac{{9.8}}{4}m$
B. $\dfrac{{9.8}}{8}m$
C. $9.8m$
D. $19.6m$

Answer 1

Hint: when a gun is aimed at a target generally it would be aimed horizontally. A bullet would come out of the barrel with some velocity horizontally. But due to gravity the bullet travels vertically too. So the bullet will have both horizontal and vertical motion. Time taken to hit the target is given and from that we will find out distance bullets travelled vertically by using kinematic formula.

Formula used: $s = ut + \dfrac{1}{2}a{t^2}$

Complete step by step answer:
let the time taken to hit the target be ‘t’ seconds.
Bullet will be having initial horizontal velocity only and it doesn’t have initial vertical velocity.
Now from the formula
$s = ut + \dfrac{1}{2}a{t^2}$ … eq(1)
Where in this problem
S = vertical distance travelled by bullet
u = initial vertical velocity of bullet
t = time taken for bullet to hit the target
a = bullet acceleration in vertical direction
According to problem u = 0, t = 0.5 sec, a = acceleration due to gravity(g) = 9.8 m/$s^2$
Substituting the above values in equation 1 we get
$s = ut + \dfrac{1}{2}a{t^2}$
$\eqalign{
  & s = 0 \times t + \dfrac{1}{2} \times 9.8 \times {[\dfrac{1}{2}]^2} \cr
  & s = \dfrac{1}{2} \times 9.8 \times {[\dfrac{1}{2}]^2} = \dfrac{{9.8}}{8}m \cr
  & \cr} $

So, the correct answer is “Option B”.

Additional Information: we can find out the vertical component of bullet velocity when it has hit the target by using kinematic formula. Acceleration due to gravity is responsible for this velocity and work-energy theorem also can be used to find this velocity.

Note: In this case the bullet is projected from the barrel horizontally. If a bullet is projected diagonally then it will have an initial y component of velocity too. Hence in formula ‘u’ will not be equal to zero. This is a particular condition in which the aim is at horizontal level to the barrel of the gun.