To calculate when the ball hits the elevator floor, we need to take into account the motion of both the ball and the elevator. Here's the step-by-step process:

Step 1: Establish the motion of the ball

We are given that the ball is thrown upwards with an initial speed $v_0 = 13.7 \, \text{m/s}$ from a height of $h_0 = 1.85 \, \text{m}$ above the elevator floor.

The motion of the ball follows the equations of uniformly accelerated motion, where the acceleration is due to gravity: $y_b(t) = h_0 + v_0 t - \frac{1}{2} g t^2$ where:

• $y_b(t)$ is the height of the ball relative to the elevator floor after time $t$,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $h_0 = 1.85 \, \text{m}$,
• $v_0 = 13.7 \, \text{m/s}$.

Thus, the equation for the ball's height above the elevator floor becomes: $y_b(t) = 1.85 + 13.7t - 4.9t^2$

Step 2: Establish the motion of the elevator

The elevator moves downwards with a constant speed of $v_e = 0.815 \, \text{m/s}$. So, the position of the elevator floor after time $t$ is: $y_e(t) = -0.815 t$

Step 3: Calculate when the ball hits the elevator floor

The ball hits the elevator floor when their positions are equal, i.e., $y_b(t) = y_e(t)$. Thus, we set: $1.85 + 13.7t - 4.9t^2 = -0.815t$ Simplify this equation: $1.85 + 13.7t - 4.9t^2 + 0.815t = 0$ $1.85 + 14.515t - 4.9t^2 = 0$ Now, rearrange it into a standard quadratic form: $-4.9t^2 + 14.515t + 1.85 = 0$

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = -4.9$, $b = 14.515$, and $c = 1.85$.

Now let's compute the solution.The solutions for the time are:

$t_1 \approx 3.13 \, \text{seconds}$ $t_2 \approx -0.12 \, \text{seconds}$

Since time cannot be negative, the valid solution is:

$t = 3.13 \, \text{seconds}$

Thus, the ball hits the elevator floor approximately 3.13 seconds after it is thrown.

Would you like further details on any step? Here are five questions that could expand on this topic:

1. How is the motion of the ball described using kinematic equations?
2. What is the effect of gravity on the ball's motion after it's thrown upwards?
3. How does the elevator's constant downward speed influence the time calculation?
4. What are the steps involved in solving quadratic equations in physics problems?
5. How would the time change if the elevator was moving upwards instead of downwards?

Tip: When solving physics problems with multiple objects in motion, always set up their individual equations first and then solve for the point where they meet or interact.