Let's solve this step by step.

1. Analyzing the problem and provided values

We need to determine:

1. Altitude $y_9$ at position 9 of the trajectory.
2. Horizontal distance $x_7$ at position 7.

Given:

• Initial velocity $v_0 = 500 \, \text{m/s}$
• Launch angle $\theta = 30^\circ$
• Acceleration due to gravity $g = 10 \, \text{m/s}^2$
• Time step between points: $\Delta t = 3.0 \, \text{s}$

2. Horizontal and vertical components of velocity

Using trigonometric decomposition:

• $v_{0x} = v_0 \cos \theta = 500 \cos 30^\circ$
• $v_{0y} = v_0 \sin \theta = 500 \sin 30^\circ$

From known trigonometric values:

• $\cos 30^\circ = \sqrt{3} / 2 \approx 0.866$
• $\sin 30^\circ = 1/2 = 0.5$

Thus: $v_{0x} = 500 \times 0.866 = 433 \, \text{m/s}, \quad v_{0y} = 500 \times 0.5 = 250 \, \text{m/s}.$

3. Altitude $y_9$ at position 9

The vertical position $y$ at any time $t$ is given by: $y = v_{0y} t - \frac{1}{2} g t^2.$

At position 9, the time is: $t_9 = 9 \Delta t = 9 \times 3.0 = 27.0 \, \text{s}.$

Substitute $v_{0y} = 250 \, \text{m/s}$, $g = 10 \, \text{m/s}^2$, and $t = 27.0 \, \text{s}$: $y_9 = 250 (27) - \frac{1}{2} (10) (27)^2.$

Simplify: $y_9 = 6750 - 5 (729).$ $y_9 = 6750 - 3645 = 3105 \, \text{m}.$

Thus: $\boxed{y_9 = 3105 \, \text{m}}$

4. Horizontal distance $x_7$ at position 7

The horizontal position $x$ at any time $t$ is given by: $x = v_{0x} t.$

At position 7, the time is: $t_7 = 7 \Delta t = 7 \times 3.0 = 21.0 \, \text{s}.$

Substitute $v_{0x} = 433 \, \text{m/s}$ and $t = 21.0 \, \text{s}$: $x_7 = 433 (21).$ $x_7 = 9093 \, \text{m}.$

Thus: $\boxed{x_7 = 9093 \, \text{m}}$

Final Answers

1. Altitude at position 9: $y_9 = 3105 \, \text{m}$
2. Horizontal distance at position 7: $x_7 = 9093 \, \text{m}$

Let me know if you'd like further clarification or additional details.

Related Questions:

1. What is the total time of flight for this projectile?
2. What is the maximum height achieved during the motion?
3. What is the horizontal range of the projectile?
4. At what time does the projectile reach the maximum height?
5. What is the velocity of the projectile at position 7?

Tip:

Break projectile motion into horizontal and vertical components for simpler calculations. Use kinematic equations for each direction independently.