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This problem involves projectile motion, asking us to determine when the velocity and displacement vectors of a particle are perpendicular.

Given:
- The initial velocity \( v_0 = 10 \, \text{m/s} \),
- The angle \( \alpha \) is such that \( \sin(\alpha) = \frac{2\sqrt{2}}{3} \),
- Gravitational acceleration \( g = 10 \, \text{m/s}^2 \).

### Step 1: Decompose the velocity
The initial velocity has two components:
- Horizontal velocity: \( v_{0x} = v_0 \cos(\alpha) \),
- Vertical velocity: \( v_{0y} = v_0 \sin(\alpha) \).

Given \( \sin(\alpha) = \frac{2\sqrt{2}}{3} \), we can calculate \( \cos(\alpha) \) using the identity \( \sin^2(\alpha) + \cos^2(\alpha) = 1 \).

First, let's compute \( \cos(\alpha) \):
\[
\sin(\alpha) = \frac{2\sqrt{2}}{3}, \quad \cos(\alpha) = \sqrt{1 - \sin^2(\alpha)}.
\]

### Step 2: Condition for perpendicularity
The velocity vector and displacement vector are perpendicular when their dot product equals zero:
\[
\vec{v} \cdot \vec{r} = 0.
\]

Let me calculate the missing values.We have:
- \( \sin(\alpha) = 0.9428 \),
- \( \cos(\alpha) = 0.3333 \).

### Step 3: Velocity components
Now, the initial velocity components are:
- \( v_{0x} = v_0 \cos(\alpha) = 10 \times 0.3333 = 3.333 \, \text{m/s} \),
- \( v_{0y} = v_0 \sin(\alpha) = 10 \times 0.9428 = 9.428 \, \text{m/s} \).

### Step 4: Projectile motion equations
The position of the particle at any time \( t \) is given by:
- Horizontal displacement: \( x(t) = v_{0x} t \),
- Vertical displacement: \( y(t) = v_{0y} t - \frac{1}{2} g t^2 \).

The velocity components at any time \( t \) are:
- Horizontal velocity: \( v_x(t) = v_{0x} \),
- Vertical velocity: \( v_y(t) = v_{0y} - g t \).

### Step 5: Condition for perpendicularity
The condition for perpendicularity is:
\[
v_x x + v_y y = 0.
\]

Substitute the expressions for \( x(t) \), \( y(t) \), \( v_x(t) \), and \( v_y(t) \), and solve for \( t \). Let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you have further questions or want a different explanation, feel free to ask!

Here are five related questions:
1. How do you calculate the time of flight for a projectile?
2. What is the significance of perpendicularity between velocity and displacement in motion?
3. How do you calculate the range of a projectile?
4. What is the formula for the maximum height reached by a projectile?
5. How does the angle of launch affect the trajectory of a projectile?

**Tip:** When solving projectile motion problems, it’s helpful to keep the horizontal and vertical components separate to simplify calculations.

Para la partícula que se lanza como se muestra, determine en qué instante la velocidad y el desplazamiento son perpendiculares.

Solve the problem of determining the time when velocity and displacement are perpendicular in projectile motion. Using key kinematic equations and trigonometric identities, this step-by-step solution helps students in Grades 10-12 understand the conditions for perpendicularity.

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