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The image you've uploaded shows problems from WebAssign related to physics and calculus, specifically dealing with kinematic equations and free-fall motion under gravity.

Let's break down the main parts of the two problems I can see:

### Problem 1 (2.2.096.EP):
A ball is thrown downward from the top of a building with an initial velocity of \( v_0 = -18 \, \text{ft/s} \) from a height \( h = 417 \, \text{ft} \).
- The equation for the height \( s(t) \) of the ball at any time \( t \) is:
  \[
  s(t) = -16t^2 + v_0 t + h
  \]
  Substituting the known values:
  \[
  s(t) = -16t^2 - 18t + 417
  \]

The questions are:
1. How long does it take the ball to reach the ground?
   To solve for the time when the ball reaches the ground, set \( s(t) = 0 \) and solve the quadratic equation:
   \[
   -16t^2 - 18t + 417 = 0
   \]

2. With what speed does the ball hit the ground?
   After finding the time \( t \) when the ball hits the ground, substitute it into the velocity equation \( v(t) = -32t + v_0 \) to get the final speed.

3. How many seconds does it take to reach 322 feet?
   Substitute \( s(t) = 322 \) and solve the quadratic equation:
   \[
   -16t^2 - 18t + 417 = 322
   \]

### Problem 2 (2.2.097):
A projectile is shot upward from the surface of the Earth with an initial velocity of \( 62 \, \text{m/s} \).
- The height \( s(t) \) of the projectile at any time \( t \) is:
  \[
  s(t) = -4.9t^2 + v_0 t + h
  \]
  Here, \( v_0 = 62 \, \text{m/s} \) and \( h = 0 \, \text{m} \), so:
  \[
  s(t) = -4.9t^2 + 62t
  \]

The questions are:
1. When does the projectile reach its maximum height?
   The time to reach the maximum height is found by solving for \( t \) when \( v(t) = 0 \). Since \( v(t) = -9.8t + v_0 \), solve \( -9.8t + 62 = 0 \).

2. What is the maximum height?
   Substitute the time \( t \) found above into \( s(t) \) to find the maximum height.

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Would you like me to solve any of these problems in detail? Also, here are five related questions you might want to explore further:

1. How does air resistance affect the motion of a falling object?
2. What is the difference between free-fall motion and projectile motion?
3. How would the calculations change if the initial velocity is upward instead of downward?
4. Can we solve these problems using numerical methods instead of analytical?
5. How does gravity differ on different planets, and how would that affect the equations?

**Tip**: When solving quadratic equations, always check if the discriminant is positive to ensure there are real solutions for time.

Let me know if you'd like to dive deeper into any specific part!

A ball is thrown downward from a 417-ft building with an initial velocity of -18 ft/s. How long does it take to reach the ground? What is the speed at impact? How long to fall to 322 feet?

Learn how to calculate the time and speed for a falling object in free-fall from a height of 417 feet with an initial downward velocity. This detailed guide explores kinematic equations and solving quadratic equations. Suitable for advanced high school students.

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