A ball is thrown vertically upward with an initial velocity of $$150$$ m s$$^{-1}$$. The ratio of velocity after 3 s and 5 s is $$\frac{x+1}{x}$$. The value of $$x$$ is _____. {take, $$g = 10$$ m s$$^{-2}$$}

A ball is thrown vertically upward with an initial speed of 150 m/s, and we are asked to determine the value of x such that the ratio of its velocities at t = 3 s and t = 5 s equals $$\frac{x+1}{x}$$.

Using the relation $$v = u - gt$$ with g = 10 m/s2 and upward taken as positive, we find the velocity at t = 3 s and t = 5 s as follows:

$$

v_3 = 150 - 10(3) = 150 - 30 = 120\;\text{m/s}

$$

$$

v_5 = 150 - 10(5) = 150 - 50 = 100\;\text{m/s}

$$

Since the required ratio is $$\frac{v_3}{v_5} = \frac{120}{100} = \frac{6}{5}$$, setting this equal to $$\frac{x+1}{x}$$ leads to

$$

\frac{x+1}{x} = \frac{6}{5}

$$

$$

5(x+1) = 6x \implies 5x + 5 = 6x \implies x = 5

$$

Hence, the correct answer is Option 4: 5.