To find the time, $\Delta t$, it takes for a ball to fall from a table of height $h$ with an initial velocity $v_i$ at an angle $\theta$ to the horizontal, and under the influence of vertical acceleration $a_y = -g$, we will focus on the vertical motion of the ball.

Step-by-step solution:

1. Vertical component of the initial velocity:
   The initial velocity can be broken into two components: $v_{i_y} = v_i \sin \theta$ This is the vertical component of the initial velocity.

2. Equation of vertical motion:
   The general equation for the vertical position of the ball as a function of time is: $y(t) = y_0 + v_{i_y} t + \frac{1}{2} a_y t^2$ Here, $y_0 = h$ is the initial height of the ball, and the final vertical position is $y(t) = 0$ (since the ball reaches the floor).

   Substituting these values into the equation gives: $0 = h + v_i \sin \theta \cdot t - \frac{1}{2} g t^2$

3. Solving for time $\Delta t$:
   We now solve this quadratic equation for $t$: $0 = h + v_i \sin \theta \cdot t - \frac{1}{2} g t^2$ Rearranging: $\frac{1}{2} g t^2 - v_i \sin \theta \cdot t - h = 0$ This is a quadratic equation of the form $at^2 + bt + c = 0$, where:

   • $a = \frac{1}{2} g$
   • $b = -v_i \sin \theta$
   • $c = -h$

   Using the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substituting the values of $a$, $b$, and $c$: $t = \frac{-(-v_i \sin \theta) \pm \sqrt{(-v_i \sin \theta)^2 - 4 \cdot \frac{1}{2} g \cdot (-h)}}{2 \cdot \frac{1}{2} g}$ Simplifying the expression: $t = \frac{v_i \sin \theta \pm \sqrt{(v_i \sin \theta)^2 + 2gh}}{g}$

4. Choosing the positive root:
   Since time $t$ must be positive, we take the positive root: $\Delta t = \frac{v_i \sin \theta + \sqrt{(v_i \sin \theta)^2 + 2gh}}{g}$

Thus, the time $\Delta t$ it takes for the ball to fall to the floor is: $\boxed{\Delta t = \frac{v_i \sin \theta + \sqrt{(v_i \sin \theta)^2 + 2gh}}{g}}$

Would you like more details on any step or have questions?

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Here are 5 related questions to explore further:

1. How would the time change if the angle $\theta$ is increased?
2. What is the effect of height $h$ on the time of flight?
3. How would you find the horizontal distance the ball travels during this time?
4. What happens to the time if the initial velocity $v_i$ is zero?
5. How would air resistance affect this motion?

Tip: Always break down a 2D motion problem into vertical and horizontal components.