Question

A projectile is shot at an angle of $\dfrac{\pi }{3}$ and a velocity of $6\dfrac{m}{s}$ . How far away will the projectile land?

Answer 1

In order to answer the above question, we will be using the concept of projectile motion. We will be calculating the horizontal range of the projectile using some simple laws of kinematics.

Complete step by step answer:
From the information provided above, we have to calculate the horizontal range of the projectile to answer the above question.

First of all, the displacement in the vertical direction for a projectile will be zero and the component of velocity in the Y direction will be $6\sin \theta$ . Hence, the equation
$s=ut+\dfrac{1}{2}a{{t}^{2}}$
Will become
$\begin{aligned} & 0=(6\sin \theta )t-\dfrac{1}{2}g{{t}^{2}} \\\ & \Rightarrow g{{t}^{2}}=2(6\sin \theta )t \\\ & \Rightarrow t=\dfrac{12\sin \theta }{g} \\\ \end{aligned}$
The above equation is the time of flight of the projectile after which the projectile lands again on the ground.
Now, the horizontal range for the projectile is the maximum displacement in the positive X direction and is given by
$Horizontal\text{ }Range\text{ }(R)=velocity(x)\times time$
$\begin{aligned} & \Rightarrow R=6\cos \theta \times \dfrac{12\sin \theta }{g} \\\ & \Rightarrow R=\dfrac{72\cos \theta \sin \theta }{g} \\\ \end{aligned}$
Placing the values of angle and gravitational acceleration, we get
$\begin{aligned} & \Rightarrow R=\dfrac{72\cos \dfrac{\pi }{3}\sin \dfrac{\pi }{3}}{9.8} \\\ & \Rightarrow R=3.18m \\\ \end{aligned}$
Therefore, the projectile will land at a distance of $3.18m$.

Additional Information:
What do we mean by the word projectile motion? Any object thrown into space with only gravity acting on it is referred to as a projectile. Gravity is the main force acting on a projectile. This doesn't mean that other forces don't affect it; it just means that their impact is minor in comparison to gravity. A trajectory is the direction that a projectile follows.
The point from which the projectile is projected is known as the point of projection and, and the angle with which the projected projectile is projected with respect to the point of projection is known as the angle of projection. After the projectile is projected, the projectile will travel in a curved path for a certain distance and then fall on to the ground. The horizontal distance travelled by the projectile on the ground is known as the horizontal range or the range of the projectile. And lastly, the total time taken by the projectile to travel this distance is known as the time of flight.

Note:
It is very important to note here that the direction of gravitational acceleration is taken to be negative while calculating the time of flight for the projectile as the projectile moves in the opposite direction of gravity in Y direction.