Use the formula 
h = vt − 16t2,
 where h is the height in feet that an object will attain (neglecting air resistance) in t seconds, and v is the initial velocity in feet per second.

A golf ball is thrown onto a cement surface and rebounds straight up. The initial velocity of the rebound is 80 ft/s. How many seconds later will the golf ball return to the ground?

The formula provided is:

\[
h = vt - 16t^2
\]

Where:
- \( h \) is the height in feet,
- \( v \) is the initial velocity in feet per second,
- \( t \) is the time in seconds,
- The constant \( 16 \) accounts for the gravitational pull (half of the acceleration due to gravity, 32 ft/s²).

In this problem, the initial velocity \( v = 80 \) ft/s, and we want to find when the golf ball returns to the ground, meaning the height \( h = 0 \).

### Step-by-step solution:

1. **Set the height equal to zero** because the ball returns to the ground when \( h = 0 \):
   \[
   0 = 80t - 16t^2
   \]

2. **Rearrange the equation** to form a standard quadratic equation:
   \[
   16t^2 - 80t = 0
   \]

3. **Factor the equation**:
   \[
   16t(t - 5) = 0
   \]

4. **Solve for \( t \)** by setting each factor equal to zero:
   \[
   16t = 0 \quad \text{or} \quad t - 5 = 0
   \]
   \[
   t = 0 \quad \text{or} \quad t = 5
   \]

The solution \( t = 0 \) corresponds to the moment the ball is initially thrown. The solution \( t = 5 \) means that the ball will return to the ground 5 seconds later.

### Final Answer:
The golf ball will return to the ground 5 seconds after it rebounds.

Would you like any further details or have any other questions?

Here are some related questions:
1. How does the equation change if we include air resistance?
2. What would happen if the initial velocity were doubled?
3. How would the equation change if the ball were thrown on a different planet with a different gravity?
4. What is the maximum height the golf ball reaches in this scenario?
5. How does the time change if the initial velocity is halved?

**Tip:** When solving quadratic equations, factoring is often the simplest method, but the quadratic formula can also be used when factoring isn't easy!

A golf ball is thrown onto a cement surface and rebounds straight up. The initial velocity of the rebound is 80 ft/s. How many seconds later will the golf ball return to the ground?

Learn how to determine when a golf ball returns to the ground after rebounding with an initial velocity of 80 ft/s using the quadratic equation h = vt − 16t². This problem involves key algebraic concepts and the laws of motion. Suitable for students in Grades 9-10.

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