A ball is tossed into the air at 2020 feet per second from a height of 77 feet. How long will it take the ball to reach the​ ground? Use the formula sequals=negative 16 t squared plus v 0 t plus s 0−16t2+v0t+s0​, where v 0 equals initialv0=initial ​velocity, s 0s0equals=initial ​altitude, and s equals altitudes=altitude in feet after t seconds.

To determine how long it will take for the ball to reach the ground, we can use the given formula:

\[
s(t) = -16t^2 + v_0t + s_0
\]

where:
- \(v_0\) is the initial velocity (2020 feet per second),
- \(s_0\) is the initial altitude (77 feet),
- \(s(t)\) is the altitude at time \(t\) (which will be 0 when the ball hits the ground).

We need to solve for \(t\) when \(s(t) = 0\).

The equation becomes:

\[
0 = -16t^2 + 2020t + 77
\]

This is a quadratic equation in the form \(at^2 + bt + c = 0\), where \(a = -16\), \(b = 2020\), and \(c = 77\). We can solve this quadratic equation using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Plugging in the values:

\[
t = \frac{-2020 \pm \sqrt{2020^2 - 4(-16)(77)}}{2(-16)}
\]

Let's calculate this step by step.

1. Calculate the discriminant:
\[
\text{Discriminant} = 2020^2 - 4(-16)(77)
\]

2. Take the square root of the discriminant.

3. Use the quadratic formula to solve for \(t\).

Let me compute this.The solutions for the time \(t\) are approximately:

\[
t_1 \approx -0.038 \text{ seconds}
\]
\[
t_2 \approx 126.288 \text{ seconds}
\]

Since time cannot be negative, we discard \(t_1\). Therefore, the ball will take approximately **126.29 seconds** to reach the ground.

Would you like more details or have any questions?

Here are 5 related questions:
1. What would be the maximum height reached by the ball?
2. How would the time change if the initial velocity were doubled?
3. What if the ball were thrown from a higher altitude?
4. How would air resistance affect the time to reach the ground?
5. Can you determine the velocity of the ball just before it hits the ground?

**Tip:** When solving quadratic equations, always consider the physical context to determine which solution is meaningful.

Learn how to calculate the time it takes for a ball tossed at 2020 feet per second from a height of 77 feet to reach the ground. This physics problem involves using the quadratic equation formula and kinematic principles. Suitable for advanced high school students.

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