A ball is dropped with a velocity v0 and an angle alpha with the vertical onto the top step of a flight of stairs consisting of 10 steps. The ball rebounds and bounces down the steps where the coefficient of restitution between ball and step is e=0.6. The friction between ball and stairs is negligible.

Use values of v0 from 1.5 m/s to 3.0 m/s in 0.5-m/s increments, and values of alpha from 18 degrees to 26 degrees in 2-degree increments. By computational programming, determine
(a) if the ball bounces down the steps without skipping any step
(b) if the ball bounces down the steps without bouncing twice on the same step.

***Need the code and the work to go along with it too.

well, you can start with the observation that the horizontal velocity is constant at v0 * sinθ

Now just figure the time needed for each hop, and see if the ball travels enough to skip a step or hit it twice.

Of course, you will need to have data on the height and width of the steps, right?

To solve this problem computationally, you can use a programming language like Python. Here's an example code to determine if the ball bounces down the steps without skipping any step or bouncing twice on the same step:

```python
import math

def calculate_height(v0, alpha, num_steps):
vy0 = v0 * math.sin(math.radians(alpha))
g = 9.8 # acceleration due to gravity
h = (vy0 ** 2) / (2 * g)
return h

def simulate_bouncing(v0, alpha, num_steps):
h_step = 0.15 # height of each step
h = calculate_height(v0, alpha, num_steps)
h_remaining = h # initial height remaining
current_step = 0

while h_remaining > 0:
if h_remaining < h_step:
return False # Ball will not reach the next step
else:
h_remaining -= h_step
current_step += 1
if current_step == num_steps:
return True # Ball reached the bottom step without skipping any step
return False

def simulate_bouncing_twice(v0, alpha, num_steps):
h_step = 0.15 # height of each step
h = calculate_height(v0, alpha, num_steps)
h_remaining = h # initial height remaining
current_step = 0

while h_remaining > 0:
if h_remaining < h_step:
return False # Ball will not reach the next step
else:
h_remaining -= h_step
current_step += 1
if current_step == num_steps:
return True # Ball reached the bottom step without bouncing twice on the same step
if h_remaining < h_step:
return False # Ball will bounce twice on the same step
return False

# Main code:
num_steps = 10 # total number of steps

# Loop for different values of v0 and alpha
for v0 in range(15, 31, 5):
for alpha in range(18, 27, 2):
v0 = v0 / 10 # convert to m/s
print("v0:", v0, "m/s, alpha:", alpha, "degrees")

# Check if the ball bounces down the steps without skipping any step
if simulate_bouncing(v0, alpha, num_steps):
print("The ball bounces down the steps without skipping any step.")
else:
print("The ball skips some step.")

# Check if the ball bounces down the steps without bouncing twice on the same step
if simulate_bouncing_twice(v0, alpha, num_steps):
print("The ball bounces down the steps without bouncing twice on the same step.")
else:
print("The ball bounces twice on the same step.")

print("------------------------")
```

This code calculates the height of the ball's initial drop using the given velocity and angle. It then simulates the bouncing motion step by step until the ball reaches the bottom or an invalid bouncing condition occurs. Finally, it checks if the ball bounced down the steps without skipping any step or bouncing twice on the same step. The code then loops through different values of `v0` and `alpha` as per the given increments.