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

The equation of trajectory is $$y = 5x(1 - x)$$, which can be rewritten as:

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

We start by comparing this with the standard equation of projectile motion:

$$y = x\tan\theta - \dfrac{g x^2}{2u^2\cos^2\theta}$$

Comparing coefficients gives:

$$\tan\theta = 5 \quad \text{...(i)}$$

$$\dfrac{g}{2u^2\cos^2\theta} = 5 \quad \text{...(ii)}$$

Next, we find the initial velocity components. Since the horizontal component of velocity is $$\hat{i}$$, we have:

$$u_x = u\cos\theta = 1 \text{ m/s}$$

Substituting into equation (ii) yields:

$$\dfrac{10}{2 \times (1)^2} = 5$$

This confirms consistency. ✔

Now, the y-component follows from:

$$u_y = u\sin\theta = u\cos\theta \times \tan\theta = 1 \times 5 = 5 \text{ m/s}$$

Therefore, the y-component of initial velocity is 5 $$\hat{j}$$.