If the initial velocity in horizontal direction of a projectile is unit vector $\hat{i}$ and the equation of trajectory is $y=5 x(1-x)$. The $y$ component vector of the initial velocity is ______________ $\hat{j}$. ($\mathrm{Take}$ $\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

<p>If the initial velocity in the horizontal direction of a projectile is represented by the unit vector $\hat{i}$ and the equation of the trajectory is given by $y = 5x(1 - x)$, we need to find the $y$ component vector of the initial velocity. (Given: $g = 10 \mathrm{\ m/s^2}$)</p> <p>The trajectory equation can be expanded as:</p> <p>$y = 5x - 5x^2$</p> <p>In the general form of a projectile's trajectory: $y = x \tan \theta - \frac{1}{2} \frac{g x^2}{v_0^2}$</p> <p>Here, the equation compares as follows:</p> <p>$\tan \theta = 5 = \frac{u_y}{u_x}$</p> <p>Given that the initial horizontal velocity component, $u_x$, is 1 (unit vector $\hat{i}$), we can find $u_y$ from the relationship:</p> <p>$u_y = 5 \times 1 = 5$</p> <p>Therefore, the $y$ component vector of the initial velocity is 5$\hat{j}$.</p>