The problem presented describes a ball rolling off a tabletop and involves two parts:

Part (a): Calculate the time the ball is in the air.

The time $t$ it takes for the ball to fall to the ground can be calculated using the equation for the vertical motion: $y = \frac{1}{2} g t^2$ where $y = 1.16$ m (height of the table) and $g \approx 9.81$ m/s² (acceleration due to gravity). Rearranging for $t$, we have: $t = \sqrt{\frac{2y}{g}}$

Part (b): Calculate the initial horizontal velocity.

The horizontal velocity $v_x$ remains constant. Using the time from part (a) and the horizontal distance $x = 1.63$ m: $x = v_x t$ $v_x = \frac{x}{t}$

Let's calculate the values for parts (a) and (b).

pythonCopy code
import math

# Constants
g = 9.81  # acceleration due to gravity in m/s^2
y = 1.16  # height of the table in meters
x = 1.63  # horizontal distance in meters

# Part (a): Calculate the time of flight
t = math.sqrt(2 * y / g)

# Part (b): Calculate the horizontal velocity
v_x = x / t

t, v_x
```### Results

**(a)** The ball is in the air for approximately $$0.486$$ seconds.

**(b)** The speed of the ball at the instant it leaves the table is approximately $$3.35$$ m/s.

Would you like further details or have any questions about these calculations?

### Related Questions
1. How does the height from which the ball is dropped affect the time it stays in the air?
2. What would be the impact on the ball’s horizontal velocity if it landed further away, say 2 meters from the table?
3. How would the presence of air resistance affect the results?
4. If the height of the table was doubled, how would this affect the time of flight and horizontal velocity?
5. Can you apply these principles to calculate the necessary speed to jump a gap of a specific distance?

### Tip
In projectile motion problems, vertical and horizontal motions are independent of each other. This separation simplifies calculations and allows you to treat each component separately.