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The given equation is:

\[
\Delta x = v_0 t + \frac{1}{2} a t^2
\]

Where:
- \(\Delta x = 18\)
- \(t = 2\)
- \(a = -8\)
- \(v_0 = ?\) (initial velocity is unknown)

We need to solve for \(v_0\), the initial velocity.

First, substitute the given values into the equation:

\[
18 = v_0 \cdot 2 + \frac{1}{2} \cdot (-8) \cdot 2^2
\]

Simplify the equation:

\[
18 = 2v_0 + \frac{1}{2} \cdot (-8) \cdot 4
\]
\[
18 = 2v_0 - 16
\]
\[
18 + 16 = 2v_0
\]
\[
34 = 2v_0
\]

Finally, divide by 2 to isolate \(v_0\):

\[
v_0 = \frac{34}{2} = 17
\]

Thus, the initial velocity \(v_0 = 17 \, \text{m/s}\).

Would you like any additional details or clarifications on this solution?

Here are five related questions you might find helpful:
1. How would the equation change if the acceleration were positive?
2. What if the time \(t\) were doubled? How would that affect the initial velocity?
3. Can you solve the same problem using a different kinematic equation?
4. How does acceleration affect the displacement in this equation?
5. What would happen if \(a = 0\) (no acceleration)?

**Tip:** When dealing with kinematic equations, always keep track of the signs of acceleration and displacement, as they are vector quantities.

Given: Δx = 18, v0 = ?, t = 2, a = -8. Find the initial velocity using the kinematic equation Δx = v0t + 1/2at².

This guide walks through solving for the initial velocity (v0) using the kinematic equation Δx = v0t + (1/2)at². Given Δx = 18, t = 2 seconds, and acceleration a = -8 m/s², find the initial velocity step by step. Suitable for high school physics students.

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