A ball is thrown vertically upward with an initial velocity of $150 \mathrm{~m} / \mathrm{s}$. The ratio of velocity after $3 \mathrm{~s}$ and $5 \mathrm{~s}$ is $\frac{x+1}{x}$. The value of $x$ is ___________.

$\left\{\right.$ take, $\left.g=10 \mathrm{~m} / \mathrm{s}^{2}\right\}$

To solve this problem, we can use the following equation of motion for the vertical velocity at any given time $t$: <br/><br/> $v = u - gt$ <br/><br/> Where:<br/><br/> - $v$ is the final velocity at time $t$<br/><br/> - $u$ is the initial velocity (150 m/s)<br/><br/> - $g$ is the acceleration due to gravity (10 m/s²)<br/><br/> - $t$ is the time in seconds <br/><br/> First, we need to find the velocities at $t = 3 \mathrm{~s}$ and $t = 5 \mathrm{~s}$. <br/><br/> For $t = 3 \mathrm{~s}$: <br/><br/> $v_3 = 150 - (10)(3) = 150 - 30 = 120 \mathrm{~m} / \mathrm{s}$ <br/><br/> For $t = 5 \mathrm{~s}$: <br/><br/> $v_5 = 150 - (10)(5) = 150 - 50 = 100 \mathrm{~m} / \mathrm{s}$ <br/><br/> Now we need to find the ratio of these velocities: <br/><br/> $\frac{v_3}{v_5} = \frac{120}{100} = \frac{6}{5} = \frac{x + 1}{x}$ <br/><br/> Next, we can set up an equation to find the value of $x$: <br/><br/> $\frac{6}{5} = \frac{x + 1}{x}$ <br/><br/> Now, we can solve for $x$: <br/><br/> $6x = 5(x + 1)$<br/><br/> $6x = 5x + 5$<br/><br/> $x = 5$ <br/><br/> The value of $x$ is 5.