We are tasked with finding the maximum height of a rocket launched from a point 20 meters above the ground. The given parameters are:

• Initial position of the rocket: $\mathbf{r}_0 = \langle 0, 0, 20 \rangle$ meters.
• Initial velocity of the rocket: $\mathbf{v}_0 = \langle 250, 400, 500 \rangle$ meters/second.
• Crosswind acceleration acting on the rocket: $\mathbf{a}_{\text{wind}} = \langle 0, 2.5, 0 \rangle$ meters/second².
• Gravitational acceleration: $\mathbf{g} = \langle 0, 0, -9.8 \rangle$ meters/second².

Step 1: Analyze the motion in the z-direction (vertical)

The motion in the $z$-direction is influenced by the initial velocity in that direction and gravity.

The vertical position function for the motion in the $z$-direction is given by: $z(t) = z_0 + v_{z0}t - \frac{1}{2} g t^2$ Where:

• $z_0 = 20 \, \text{m}$ (initial height),
• $v_{z0} = 500 \, \text{m/s}$ (initial vertical velocity),
• $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity).

To find the time $t_{\text{max}}$ when the maximum height is reached, we set the vertical velocity to zero at the peak: $v_z(t) = v_{z0} - g t = 0$ $t_{\text{max}} = \frac{v_{z0}}{g} = \frac{500}{9.8} \approx 51.02 \, \text{seconds}.$

Step 2: Calculate the maximum height

Substitute $t_{\text{max}}$ into the equation for $z(t)$ to find the maximum height: $z_{\text{max}} = z_0 + v_{z0}t_{\text{max}} - \frac{1}{2} g t_{\text{max}}^2$ $z_{\text{max}} = 20 + 500(51.02) - \frac{1}{2}(9.8)(51.02)^2$ Now, let's compute this expression.

The maximum height of the rocket is approximately 12,775.1 meters, and it reaches this height after about 51.02 seconds.

Would you like further details or have any questions?

Here are five related questions to explore:

1. How would the maximum height change if the initial velocity in the z-direction were increased?
2. How long does it take for the rocket to fall back to the ground after reaching maximum height?
3. What is the total time of flight of the rocket?
4. How does the crosswind affect the horizontal displacement of the rocket?
5. What would happen if there were no gravity acting on the rocket?

Tip: Always analyze motion in each direction separately when multiple forces or components are involved.