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The problem you have presented describes the motion of a ball launched vertically upward with an initial velocity of \( v_0 = 20 \, \text{m/s} \), under the influence of gravity with acceleration \( a(t) = -9.8 \, \text{m/s}^2 \). The goal is to find the position function \( s(t) \) of the particle as a function of time, using integration to solve for the velocity and then the position.

### Step 1: Velocity Function \( v(t) \)

The acceleration \( a(t) = -9.8 \, \text{m/s}^2 \) is constant. To find the velocity function \( v(t) \), we integrate the acceleration:

\[
v(t) = \int a(t) \, dt = \int -9.8 \, dt = -9.8t + C_1
\]

We know that the initial velocity \( v(0) = 20 \, \text{m/s} \), so we can solve for \( C_1 \):

\[
v(0) = -9.8(0) + C_1 = 20 \implies C_1 = 20
\]

Thus, the velocity function is:

\[
v(t) = -9.8t + 20
\]

### Step 2: Position Function \( s(t) \)

Next, we find the position function \( s(t) \) by integrating the velocity function:

\[
s(t) = \int v(t) \, dt = \int (-9.8t + 20) \, dt = -4.9t^2 + 20t + C_2
\]

The initial position \( s(0) = 0 \) (since the ball is launched from the ground), so we solve for \( C_2 \):

\[
s(0) = -4.9(0)^2 + 20(0) + C_2 = 0 \implies C_2 = 0
\]

Thus, the position function is:

\[
s(t) = -4.9t^2 + 20t
\]

### Final Position Function:

The position of the ball as a function of time is:

\[
s(t) = -4.9t^2 + 20t
\]

This equation models the ball's vertical motion, taking into account the effects of gravity.

Would you like further explanation or have any other questions?

### Related Questions:

1. How do we determine the maximum height of the ball using the position function?
2. What is the time it takes for the ball to reach its highest point?
3. How can we find the total time the ball spends in the air before hitting the ground again?
4. How would the problem change if the initial velocity was negative (i.e., the ball is thrown downward)?
5. How does air resistance alter the motion described by the position function?

### Tip:

Always verify initial conditions when solving motion problems to ensure the constants of integration are accurately determined.

En un experimento, se lanza una pelota verticalmente hacia arriba desde el suelo con una velocidad inicial de 20 m/s. La aceleración de la gravedad es constante y es de -9.8 m/s². Se desea encontrar la función de posición s(t) de la pelota en función del tiempo.

Solve the problem of a ball launched vertically with an initial velocity of 20 m/s under gravity. This solution covers concepts from calculus, including velocity and position functions, with detailed integration steps. Ideal for advanced high school or undergraduate physics students.

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