A projectile is given an initial velocity of $(\hat{i} + 2\hat{j})\,\text{m/s}$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g = 10\,\text{m/s}^{2}$, the equation of its trajectory is :

Step 1: Identify the components of the initial velocity.

The initial velocity vector is $\vec{u} = \hat{i} + 2\hat{j}$ m/s, and the acceleration due to gravity is $g = 10\,\text{m/s}^2$.

$$u_x = 1\,\text{m/s}$$ $$u_y = 2\,\text{m/s}$$

Step 2: Write the equations of motion.

Express the horizontal and vertical displacements as functions of time $t$.

$$x = u_x t$$ $$x = t$$ $$y = u_y t - \frac{1}{2}gt^2$$ $$y = 2t - 5t^2$$

Step 3: Eliminate time to find the trajectory equation.

Substitute $t = x$ into the vertical displacement equation.

$$y = 2x - 5x^2$$

Answer: $y = 2x - 5x^2$